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Georgia Department of Education

Accelerated Mathematics III

Frameworks

Student Edition

Unit 1

Date Analysis

2

nd Edition

April, 2011



Accelerated Mathematics III Unit 1 2nd Edition


Dr. John D. Barge, State School Superintendent

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Table of Contents

INTRODUCTION .........................................................................................................................3

Colors of Reese’s Pieces Candies Learning Task…………...........................................................7

1. Why do I need to learn more about statistics? ..................................................7

2. Reviewing some basics ...................................................................................9

3. How many orange candies should I expect in a bag of Reese’s Pieces? ........9

4. Sampling Distribution of p̂ ............................................................................11

5. Simulating the Selection of Orange Reese’s Pieces .......................................12

6. Applying the CLT for Sample Proportions ....................................................14

Pennies Learning Task..................................................................................................................16

1. Sampling Distribution of a Sample Mean from a Normal Population ...........16

2. How Old are Your Pennies? ...........................................................................17

3. Applying the CLT for Sample Means ............................................................20

Gettysburg Address Learning Task..............................................................................................21

1. Sampling Distribution of a Sample Mean from a Normal Population ...........21

2. How Long are the Words in the Gettysburg Address? ...................................23

3. Applying the CLT for Sample Means ............................................................25

4. Supporting Documents: Gettysburg Address and Word List .........................26

Confidence Intervals Learning Tasks .........................................................................................29

1. The Empirical Rule .........................................................................................29

2. Confidence Intervals .......................................................................................29

3. President’s Approval Ratings: Part 1 ..............................................................31

4. President’s Approval Ratings: Part 2 ..............................................................32

5. Racing Hearts ..................................................................................................33

6. Synthesis of Confidence Intervals ..................................................................35





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Accelerated Mathematics III – Unit 1

Data Analysis

Student Edition INTRODUCTION:

In Accelerated Mathematics II, students began examining sampling distributions and

sampling variability, laying the groundwork for the Central Limit Theorem, a major topic in this

unit. Then, students looked at a number of discrete probability distributions and at one particular

continuous probability distribution, the normal distribution. This unit will continue linking the

two. The Central Limit Theorem allows us to use a normal distribution to approximate the

distribution of sample means and proportions, even when the original sampling distribution itself

is not a normal distribution. The second major topic is constructing confidence intervals and

understanding margin of error. These ideas are prevalent in the media and, therefore, important

for all students to understand.

Much of the content in this unit is beyond what was traditionally taught in Georgia and may

be beyond the mathematical and statistical training received by many teachers. Some general

references that may be helpful to teachers are the Guidelines for Assessment and Instruction in

Statistics Education (GAISE) Report, which can be found online at

www.amstat.org/education/gaise, NCTM’s Navigating through Data Analysis in Grades 9 – 12,

and “Statistics in the High School Mathematics Curriculum: Building Sound Reasoning under

Uncertain Conditions” by Richard Scheaffer and Josh Tabor in the August 2008 Mathematics

Teacher. Also, the school’s AP Statistics teacher or local RESA math specialist may be able to

provide assistance.

It is assumed throughout the unit that students already know how to use normal distribution

tables or how to calculate normal probabilities on their calculators. These skills were taught in

GPS Accelerated Pre-Calculus and should be maintained in the present unit. Other previous

topics, e.g. sampling distributions, will be reviewed throughout the unit.

ENDURING UNDERSTANDINGS:

As the sample size increases, the value of a statistic approaches the true value of the

population parameter, the standard deviation of the sample means is n

, and the

standard deviation of the sample proportions is 1p p

n. These formulas for the

standard deviation are valid as long as the population is at least 10 times the size of the

sample.

http://www.amstat.org/education/gaise





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The Central Limit Theorem allows us to use normal probability calculations, given

certain conditions are met, for sample means and proportions even if the original

distributions are non-normal.

Confidence intervals provide a range of values that estimate the population parameter.

The margin of error, often cited in media articles, tells how accurate we believe our

estimate of the parameter to be. To decrease the margin of error, we could increase the

sample size or decrease how confident we need the result to be.

KEY STANDARDS ADDRESSED:

MA3D1. Using simulation, students will develop the idea of the central limit theorem.

MA3D2. Using student-generated data from random samples of at least 30 members,

students will determine the margin of error and confidence interval for a specified level of

confidence.

MA3D3. Students will use confidence intervals and margin of error to make inferences

from data about a population. Technology is used to evaluate confidence intervals, but

students will be aware of the ideas involved.

RELATED STANDARDS ADDRESSED:

MA3P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts.

c. Apply and adapt a variety of appropriate strategies to solve problems.

d. Monitor and reflect on the process of mathematical problem solving.

MA3P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics.

b. Make and investigate mathematical conjectures.

c. Develop and evaluate mathematical arguments and proofs.

d. Select and use various types of reasoning and methods of proof.

MA3P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication.

b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and

others.

c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely.





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MA3P4. Students will make connections among mathematical ideas and to other

disciplines. a. Recognize and use connections among mathematical ideas.

b. Understand how mathematical ideas interconnect and build on one another to produce a

coherent whole.

c. Recognize and apply mathematics in contexts outside of mathematics.

MA3P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas.

b. Select, apply, and translate among mathematical representations to solve problems.

c. Use representations to model and interpret physical, social, and mathematical

phenomena.

UNIT OVERVIEW:

The unit begins with students reading and discussing an internet article on presidential approval

ratings, employing percents and margins of error. This motivates the unit and will be revisited in

the Confidence Intervals Tasks.

The Reese’s Pieces Candies task reviews ideas about sampling distributions of sample

proportions and develops this knowledge into the Central Limit Theorem (CLT) for Proportions.

In addition to collecting data with actual candy, students will use simulation to fully develop the

CLT. The task concludes with a problem intended to synthesize the ideas from the simulation.

The next two tasks, Pennies and Gettysburg Address, are nearly identical. Both tasks review

sampling distributions of sample means from normal populations, a topic from Acc. Math II.

Then, either with pennies or words from the Gettysburg Address, students will draw samples of

various sizes, create whole class plots, and discuss the results. This activity leads to the CLT for

Means. The tasks conclude with a problem intended to synthesize the ideas from the activity.

The final series of learning tasks builds on students’ understanding of the empirical rule to

develop confidence intervals and margin of error. The article from the beginning of the unit will

serve as a catalyst and basis for the investigation of margin of error and confidence intervals for

proportions. Students will then simulate samples for the creation of additional samples in an

effort to understand what it means to say that one is 95% confident. The last part of this series

addresses confidence intervals for sample means, including the use of data collection and

simulation.

The culminating task requires students to synthesize the statistical knowledge gained throughout

high school. They must design, implement, and analyze the results of a survey or experiment,

paying particular attention to the use of statistical inference.





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VOCABULARY AND FORMULAS

Central Limit Theorem:

Choose a simple random sample of size n from any population with mean and standard

deviation . When n is large (at least 30), the sampling distribution of the sample mean

x is approximately normal with mean and standard deviation n

.

Choose a simple random sample of size n from a large population with population

parameter p having some characteristic of interest. Then the sampling distribution of the

sample proportion p̂ is approximately normal with mean p and standard deviation

1p p

n. This approximation becomes more and more accurate as the sample size n

increases, and it is generally considered valid if the population is much larger than the

sample, i.e. np 10 and n(1 – p) 10.

The CLT allows us to use normal calculations to determine probabilities about sample

proportions and sample means obtained from populations that are not normally

distributed.

Confidence Interval is an interval for a parameter, calculated from the data, usually in the form

estimate margin of error. The confidence level gives the probability that the interval will

capture the true parameter value in repeated samples.

Margin of Error is the value in the confidence interval that says how accurate we believe our

estimate of the parameter to be. The margin of error is comprised of the product of the z-score

and the standard deviation (or standard error of the estimate). The margin of error can be

decreased by increasing the sample size or decreasing the confidence level.

Parameter is a number that describes the population. A parameter is a fixed number, but in

practice we do not know its value because we cannot examine the entire population.

Sample Mean is a statistic measuring the average of the observations in the sample. It is written

as x . The mean of the population, a parameter, is written as .

Sample Proportion is a statistic indicating the proportion of successes in a particular sample. It

is written as p̂ . The population proportion, a parameter, is written as p.

Sampling Distribution of a statistics is the distribution of values taken by the statistic in all

possible samples of the same size from the same population.

Sampling Variability refers to the fact that the value of a statistic varies in repeated random

sampling.





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Statistic is a number that describes a sample. The value of the statistics is known when we have

taken a sample, but it can change from sample to sample. We often use a statistic to estimate an

unknown parameter.





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COLORS OF REESE’S PIECES CANDIES1 LEARNING TASK

1. Why do I need to learn more about statistics?

a. Read the following article. What do the numbers in the article represent?

b. How reliable do you think the ratings are?

c. How do you think pollsters determine approval ratings such as these?

d. What do you think a margin of error is? Why is that important?

During this unit, you will learn the answers to these questions and how those answers are

important.

Excerpt from http://www.cnn.com/2005/POLITICS/12/19/bush.poll/

Poll: Iraq speeches, election don't help Bush

Tuesday, December 20, 2005; Posted: 12:56 a.m. EST (05:56 GMT)

CNN -- President Bush's approval ratings do not appear to have changed significantly, despite a number of recent speeches he's given to shore up public support for the war in Iraq and its historic elections on Thursday.

A CNN/USA Today Gallup poll conducted over the weekend found his approval rating stood at 41 percent, while more than half, or 56 percent, disapprove of how the president is handling his job. A majority, or 52 percent, say it was a mistake to send troops to Iraq, and 61 percent say they disapprove of how he is handling Iraq specifically. The margin of error was plus or minus 3 percentage points. ... The poll was nearly split, 49 percent to 47 percent, between those who thought the U.S. will

either "definitely" or "probably" win, and those who said the U.S. will lose. That said, 69 percent

of those polled expressed optimism that the U.S. can win the war. The margin of error for how

respondents assessed the war was plus or minus 4.5 percentage points.

...

Although half those polled said that a stable government in Iraq was likely within a year, 62

percent said Iraqi forces were unlikely to ensure security without U.S. assistance. And 63 percent

said Iraq was unlikely to prevent terrorists from using Iraq as a base. The margin of error on

questions pertaining to troop duration in Iraq, as well as the country's future, was plus or minus 3

percentage points.

1 This task is based on chapter 16 from Rossman, A. J., Chance, B. L., & VonOehsen, J. B. (2002). Workshop

statistics: Discovery with data and the graphing calculator (2nd

edition). Emeryville, CA: Key Curriculum Press.





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The poll interviewed 1,003 adult Americans and found that the public has also grown more

skeptical about Bush's key arguments in favor of the war. Compared with two years ago, when

57 percent considered Iraq a part of the war on terrorism, 43 percent think so now. In the

weekend poll, 55 percent said they view the war in Iraq as separate from the war on terror. The

margin of error on this line of questioning was plus or minus 3 percentage points.

On the domestic front, 56 percent of those polled say they disapprove of how Bush is handling

the economy; by contrast, 41 percent approve. The margin of error was plus or minus 3

percentage points.

The president may find support for his call to renew the Patriot Act. Forty-four percent said they

felt the Patriot Act is about right, and 18 percent said it doesn't go far enough. A third of

respondents say they believe the Patriot Act has gone too far in restricting people's civil liberties

to investigate suspected terrorism.

Nearly two-thirds said they are not willing to sacrifice civil liberties to prevent terrorism, as

compared to 49 percent saying so in 2002. The margin of error was plus or minus 4.5 percentage

points for those questions.

2. Reviewing some basics:

a. Think about a single bag of Reese’s Pieces. Does this single bag represent a sample of

Reese’s Pieces or the population of Reese’s pieces?

b. We use the term statistic to refer to measures based on samples and the term parameter to

refer to measures of the entire population. If there are 62 Reese’s Pieces in your bag, is 62 a

statistic or a parameter? If Hershey claims that 25% of all Reese’s Pieces are brown, is 25%

a statistic or a parameter?

We also use different symbols to represent statistics and parameters. The following table

will be very useful as we continue through this unit.

Parameter Statistic

Proportion P p̂ “p-hat”

Mean “mu” x “x-bar”

Standard deviation “sigma” S

Number N N

3. How many orange candies should I expect in a bag of Reese’s Pieces?

a. From your bag of Reese’s Pieces, take a random sample of 10 candies. Record the count

and proportion of each color in your sample.





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Orange Yellow Brown

Count

Proportion

b. Do you know the value of the proportion of orange candies manufactured by Hershey?

c. Do you know the value of the proportion of orange candies among the 10 that you selected?

d. Do you think that every student in the class obtained the same proportion of orange candies

in his or her sample? Why or why not?

e. Combine your results with the rest of the class and produce a dotplot for the distribution of

sample proportions of orange candies (out of a sample of 10 candies) obtained by the class

members.

f. What is the average of the sample proportions obtained by your class?

g. Put the Reese’s Pieces back in the bag and take a random sample of 25 candies. Record the

count and proportion of each color in your sample.

Orange Yellow Brown

Count

Proportion

h. Combine your results with the rest of the class and produce a dotplot for the distribution of


members. Is there more or less variability than when you sampled 10 candies? Is this what

you expected? Explain.

i. What is the average of the sample proportions (from the samples of 25) obtained by your

class? Do you think this is closer or farther from the true proportion of oranges than the

value you found in f? Explain.

j. This time, take a random sample of 40 candies. Record the count and proportion of each

color in your sample.

Orange Yellow Brown

Count

Proportion





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k. Combine your results with the rest of the class and produce a dotplot for the distribution of


members. Is there more or less variability than the previous two samples? Is this what you

expected? Explain.

l. What is the average of the sample proportions (from the samples of 40) obtained by your

class? Do you think this is closer or farther from the true proportion of oranges than the

values you found in f and i? Explain.

4. Sampling Distribution of p̂

We have been looking a number of different sampling distributions of p̂ , but we have seen that

there is great variability in the distributions. We would like to know that p̂ is a good estimate

for the true proportion of orange Reese’s Pieces. However, there are guidelines for when we can

use the statistic to estimate the parameter. This is what we will investigate in the next section.

First, however, we need to understand the center, shape, and spread of the sampling distribution

of p̂ .

We know that if we are counting the number of Reese’s pieces that are orange and comparing

with those that are not orange, then the counts of oranges follow a binomial distribution (given

that the population is much larger than our sample size).

a. Recall from Math 3 the formulas for the mean and standard deviation of a binomial

distribution.

b. Given that p̂ = X/n, where X is the count of oranges and n is the total in the sample, how

might we find p̂ and p̂ ? Find formulas for each statistic.

c. This leads to the statement of the characteristics of the sampling distribution of a sample

proportion.

The Sampling Distribution of a Sample Proportion:

Choose a simple random sample of size n from a large population with population

parameter p having some characteristic of interest. Let p̂ be the proportion of the sample

having that characteristic. Then:

o The mean of the sampling distribution is ____.

o The standard deviation of the sampling distribution is ___________.

d. Let’s look at the standard deviation a bit more. What happens to the standard deviation as

the sample size increases? Try a few examples to verify your conclusion. Then use the

formula to explain why your conjecture is true.





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If we wanted to cut the standard deviation in half, thus decreasing the variability of p̂ ,

what would we need to do in terms of our sample size?

e. Caution: We can only use the formula for the standard deviation of p̂ when the

population is at least 10 times as large as the sample.

For each of the samples taken in part 3, determine what the population of Reese’s Pieces

must be for us to use the standard deviation formula derived above. Is it safe to assume

that the population is at least as large as these amounts? Explain.

5. Simulating the Selection of Orange Reese’s Pieces

As you saw above, there is variation in the distributions depending on the size of your sample

and which sample is chosen. To better investigate the distribution of the sample proportions, we

need more samples and we need samples of larger size. We will turn to technology to help with

this sampling. For this simulation, we need to assume a value for the true proportion of orange

candies. Let’s assume p = 0.45.

a. First, let’s imagine that there are 100 students in the class and each takes a sample of 50

Reese’s Pieces. We can simulate this situation with your calculator.

Type randBin(50,0.45) in your calculator. (randBin is found in the following way:

MathPROB6.) What number did you get? Compare with a neighbor. What do you

think this command does?

How could you obtain the proportion that are orange rather than the count?

b. Now, we want to generate 100 samples of size 50. This time, input

randBin(50,0.45,100)/50L1. The latter part (store in L1) puts all of the outputs into List

1.

Using your Stat Plots, create a histogram or stem-and-leaf plot of the proportions of

orange candies. Sketch the graph below. (If you are using a program that creates dotplots,

create a dotplot instead.)

Do you notice a pattern in the distribution of the sample proportions? Explain.

c. Find the mean and standard deviation of the output using 1-Var Stats. How do these

compare with the theoretical mean and standard deviation for a sampling distribution of a

sample proportion from part 3?

Mean: ______________ Standard Deviation: ______________





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d. Use the TRACE button on the calculator to count how many of the 100 sample

proportions are within 0.07 of 0.45. Note: 0.07 is close to the standard deviation you

found above, so we are going about one standard deviation on each side of the mean.

Then repeat for within 0.14 and for within 0.21. Record the results below:

Number of the 100

Sample Proportions

Percentage of the 100

Sample Proportions

Within 0.07 of 0.45

Within 0.14 of 0.45

Within 0.21 of 0.45

e. If each of the 100 students who sampled Reese’s Pieces were to estimate the population

proportion of orange candies by going a distance of 0.14 on either side of his or her

sample proportion, what percentage of the 100 students would capture the actual

proportion (0.45) within this interval?

f. If you did not know the actual proportion of oranges, would the simulation above provide

you with a definitive way of knowing whether your sample was within 0.14 of the mean?

Explain.

g. Simulate drawing out 200 Reese’s Pieces 100 times. Find the mean and standard

deviation of the set of sample proportions in this simulation. Compare with the theoretical

mean and standard deviation of the sampling distribution with sample size 200.

h. How does the plot of the sampling distribution different from the above plot? How do the

mean and standard deviation compare?

i. What percentage of the 200 sample proportions fall within 0.07 of 0.45 (or approximately

2 standard deviations)? How does this compare with the answer to part e?

j. You should notice that these distributions follow an approximately normal distribution, a

topic you learned about in Math 2. You also learned the Empirical Rule that states how

much of the data will fall within 1, 2, and 3 standard deviations of the mean. Restate the

rule:

In a normal distribution with mean and standard deviation :

o ____% of the observations fall within 1 standard deviation (1 ) of the mean ( ).

o ____% of the observations fall within 2 standard deviations (2 ) of the mean ( ).

o ____% of the observations fall within 3 standard deviation (3 ) of the mean ( ).





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k. Do your answers to parts e and i agree with the Empirical Rule? Explain.

This leads us to an important result in statistics: the Central Limit Theorem (CLT) for a

Sample Proportion:

Choose a simple random sample of size n from a large population with population parameter p

having some characteristic of interest. Then the sampling distribution of the sample proportion

p̂ is approximately normal with mean p and standard deviation 1p p

n. This approximation

becomes more and more accurate as the sample size n increases, and it is generally considered

valid if the population is much larger than the sample, i.e. np 10 and n(1 – p) 10.

l. How might this theorem be helpful? What advantage does this theorem provide in

determining the likelihood of events?

6. Applying the CLT for Sample Proportions2

A USA Today poll asked a random sample of 1012 U.S. adults what they did with their cereal

milk after they have eaten the cereal. Of the respondents, 67% said that they drink the milk.

a. Is it possible to know for certain what percentage of U.S. adults drink their cereal milk?

Explain.

b. Suppose we know that 70% of U.S. adults drink the cereal milk. Find the mean and

standard deviation of the proportion p̂ of the sample that say they drink the cereal milk.

c. Explain why you can use the formula for the standard deviation of p̂ in this situation.

d. What sample size would be required to reduce the standard deviation of the sample

proportion to one-third the value you found in part b?

e. Check that you can use the normal approximation for the distribution of p̂ , i.e. the

Central Limit Theorem for Sample Proportions.

2 Problem adapted from Yates, D. S., Moore, D. S., & Starnes, D. S. (2003). The Practice of Statistics: TI-83/89

Graphing Calculator Enhanced (2nd

ed.). New York: W.H. Freeman.





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f. In Math 3, you learned to calculate z-scores using the following formula: X

z .

Note that X is a statistic, is a parameter for the mean, and is the parameter for the

standard deviation. More specifically, whenever we standardize values, we take the

estimate (or statistic) minus the corresponding parameter and divide the difference by the

corresponding standard deviation. We will standardize proportions in the same way.

Substitute the statistics and parameters for proportions into the z-score formula to obtain

our standardization formula for proportions.

g. Find the probability of obtaining a sample of 1012 adults in which 67% or fewer say they

drink the cereal milk. (Use the standard normal distribution.)

h. Find the probability that p̂ takes a value between 0.67 and 0.73. This will tell us if a

simple random sample of 1012 adults will usually give a result p̂ within 3 percentage

points of the true population proportion.





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PENNIES LEARNING TASK

1. Sampling Distribution of a Sample Mean from a Normal Population

The scores of individual students on the ACT entrance exam have a normal distribution with

mean 18.6 and standard deviation 5.9.

a. Use your calculator to simulate the scores of 25 randomly selected students who took the

ACT. Record the mean and standard deviations of these 25 people in the table below.

Repeat, simulating the scores of 100 people. (To do this, use the following command:

randNorm( , , n)L1.)

Mean Standard Deviation

Population 18.6 5.9

25 people

100 people

b. As a class, compile the means for the sample of 25 people. Determine the mean and

standard deviation of this set of means. That is, calculate x and x . How does the

mean of the sample means compare with the population mean? How does the standard

deviation of the sample means compare with the population standard deviation?

c. Describe the plot of this set of means. How does the plot compare with the normal

distribution?

d. As a class, compile the means for the sample of 100 people. Determine the mean and




Describe the plot of this set of means. How does the plot compare with the normal distribution?

e. Determine formulas for the mean of the sample means, x , and the standard deviation of

the sample means, x . Compare with a neighbor.

f. Just as we saw with proportions, the sample mean is an unbiased estimator of the

population mean.

The Sampling Distribution of a Sample Mean:





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Choose a simple random sample of size n from a large population with mean and

standard deviation . Then:

o The mean of the sampling distribution of x is ____.

o The standard deviation of the sampling distribution of x is ___________.

g. Again, we must be cautious about when we use the formula for the standard deviation of

x . What was the rule when we looked at proportions? It is the same here.

h. Put these latter facts together with your response to part b to complete the following

statement:

Choose a simple random sample of size n from a population that has a normal

distribution with mean and standard deviation . Then the sample mean x has a

____________distribution with mean ________ and standard deviation ____________.

.

This problem centered on a population that was known to be normally distributed. What about

populations that are not normally distributed? Can we still use the facts above? Let’s investigate!

2. How Old are Your Pennies?3

a. Make a frequency table of the year and the age of the 25 pennies you brought to class. Find

the average age of the 25 pennies. Record the mean age as 25x .

25x = ___________

Example (if the year is 2009):

Year Age Frequency

2009 0 3

2008 1 6

2007 2 3

... ... ...

3 The Workshop Statistics books provide an alternatives to actually working with pennies; however, it is not possible

to recreate the methods here. It requires assigning three-digit numbers to penny ages and using a random number

generator to “sample” pennies.





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b. Put your 25 pennies in a cup or bag and randomly select 5 pennies. Find the average age of

the 5 pennies in your sample, and record the mean age as 5x . Replace the pennies in the

cup, and repeat.

1 5x = ________________

2 5x = ________________

c. Repeat the process two more times, this time removing 10 pennies at a time. Calculate the

average age of the sample of 10 pennies and record as 10x .

1 10x = ________________

2 10x = ________________

e. Clear a space on the floor and use masking tape to make a number line (horizontal axis)

with ages marked from 0 to about 30 on the axis. Each interval should be a little more

than the width of a penny. Place the pennies on the axis according to age, making a penny

dotplot on the floor.

Look at the shape of the final dotplot. Describe the distribution of the pennies’ ages.

f. Make a second axis on the floor and label it with increments of 0.5. Each interval should

be a little more than the width of a nickel. Use the nickels to plot the means for the

sample size 5. What is the shape of the dotplot for the distribution of 5x ? How does it

compare with the original distribution of pennies’ ages?

g. Make a third axis on which to create a dotplot of the means for the sample size 10. Use

dimes for this plot. What is the shape of the dotplot for the distribution of 10x ? How

does it compare with the previous plots?

h. Finally, make a fourth axis. On this axis, use the quarters to record the means for the

sample size of 25. Describe the shape of the dotplot.

i. Divide into groups for the next part of this task. There should be at least 4 groups. Each

group will take one of the dotplots above and find the mean and standard deviation of the

sample means. (Because there are so many pennies, two groups should separately

determine the overall mean and standard deviation and check each other.)

Post a table similar to the one below on the board so that groups can post their results.

Record all groups’ result here.

Mean Standard Deviation Shape of the Distribution





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“Population”

Samples of 5

Samples of 10

Samples of 25

j. Previously, we stated that if samples were taken from a normal distribution, that the mean

and standard deviation of the sampling distribution of sample means was also normal

with x and xn

. In this activity, we did not begin with a normal distribution.

However, compare the means for the samples of 5, 10, and 25 with the overall mean of

the pennies. Then compare the standard deviations with the standard deviation of all the

pennies. Do these formulas appear to hold despite the population of penny ages being

obviously non-normal? Explain.

k. Suppose that the U.S. Department of Treasury estimated that the average age of pennies

presently in circulation is 12.25 years with a standard deviation of 9.5. Determine the

theoretical means and standard deviations for the sampling distributions of sample size 5,

10, and 25.


Population 12.25 9.5

Samples of 5

Samples of 10

Samples of 25

l. This brings us to the Central Limit Theorem (CLT) for Sample Means:

Choose a simple random sample of size n from any population, regardless of the original

shape of the distribution, with mean and finite standard deviation . When n is large,

the sampling distribution of the sample mean x is approximately normal with mean

______ and standard deviation ________.

Note: The statement “when n is large” seems a bit ambiguous. A good rule of thumb is

that the sample size should be at least 30, as we can see in the dotplots above. When the

sample sizes were sample, i.e. 5 and 10, the plots were still quite right skewed.

3. Applying the CLT for Sample Means

Whenever we have a normal distribution, we are able to use normal tables and normal

calculations to find probabilities. Due to the CLT, we can use normal calculations to determine

probabilities about sample means drawn from large samples.

a. First, we need to recall how to standardize scores, that is, find z-scores. State the formula

for z-scores that you learned in Math 3.





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Recall that whenever we standardize values, we take the estimate (or statistic) minus the

corresponding parameter and divide that difference by the corresponding standard

deviation. We will standardize sample means in the same way. Substitute the statistics

and parameters for sample means into the z-score formula to obtain our standardization

formula for sample means.

b. Consider a candy bar whose weight varies according to a normal distribution with a mean

of 2.23 ounces and a standard deviation of 0.05 ounces. Find the probability that a single

candy bar weighs less than 2.2 ounces.

c. Suppose you take a sample of 20 candy bars. What are the mean and standard deviation

of the sampling distribution of the average weight of this size sample?

d. Find the probability that the mean weight of these 20 candy bars is less than 2.2 ounces.

e. Find the probability that the mean weight of these candy bars is between 2.21 and 2.25

ounces.

f. How would you expect your answers to parts d and e would be different if the sample

size were 50 instead of 20?

g. Calculate these probabilities and comment on your conjecture.

h. Which of your answers, if any, to parts b, c, d, e, and g would be affected if the

distribution of the candy bar weights was not normally distributed? Explain.





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GETTYSBURG ADDRESS LEARNING TASK

1. Sampling Distribution of a Sample Mean from a Normal Population

The scores of individual students on the ACT entrance exam have a normal distribution with

mean 18.6 and standard deviation 5.9.

a. Use your calculator to simulate the scores of 25 randomly selected students who took the

ACT. Record the mean and standard deviations of these 25 people in the table below.

Repeat, simulating the scores of 100 people. (To do this, use the following command:

randNorm( , , n)L1.)


Population 18.6 5.9

25 people

100 people

b. As a class, compile the means for the sample of 25 people. Determine the mean and




c. Describe the plot of this set of means. How does the plot compare with the normal

distribution?

d. As a class, compile the means for the sample of 100 people. Determine the mean and




e. Describe the plot of this set of means. How does the plot compare with the normal

distribution?

f. Determine formulas for the mean of the sample means, x , and the standard deviation of

the sample means, x . Compare with a neighbor.





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Just as we saw with proportions, the sample mean is an unbiased estimator of the population

mean.

The Sampling Distribution of a Sample Mean:

Choose a simple random sample of size n from a large population with mean and

standard deviation . Then:

o The mean of the sampling distribution of x is ____.

o The standard deviation of the sampling distribution of x is ___________.

g. Again, we must be cautious about when we use the formula for the standard deviation of

x . What was the rule when we looked at proportions? It is the same here.

h. Put these latter facts together with your response to part b to complete the following

statement:

Choose a simple random sample of size n from a population that has a normal

distribution with mean and standard deviation . Then the sample mean x has a

____________distribution with mean ________ and standard deviation ____________.

This problem centered on a population that was known to be normally distributed. What about

populations that are not normally distributed? Can we still use the facts above? Let’s investigate!

2. How Long are the Words in the Gettysburg Address?

(Note: The Gettysburg Address and Word List are at the end of this task.)

a. To answer the question of how long the words in the Gettysburg address are, we want to

take a sample of the words. Propose different ways you might take a random sample of 5

words from the Gettysburg address.

b. The last page of this activity lists the words from the Gettysburg address. There are 268

words, and each word is assigned a number from 1 (001) to 268. To select a simple

random sample of 5 words, we need to generate 5 distinct random integers between 1 and

268. (Use the following command: randInt(1, 268, 5). If any of the numbers repeat,

repeat the command until you have 5 distinct integers.)

Find the words on the list that correspond to these integers. List the word lengths. Find

the average length of the 5 words in your sample, and record the mean length as 5x .

Generate a new set of 5 distinct integers, and repeat. Also record each mean length on a

separate post-it note. Make sure the post-it is labeled, e.g. 5x !





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Sample One Sample Two

Random Integers

Word Lengths

Average Length 1 5x = 2 5x =

c. Repeat the process two more times, this time choosing 10 words at a time. Calculate the

average length of the sample of 10 words and record as 10x . Also record each mean

length on a separate post-it note. Make sure the post-it is labeled!

Sample One Sample Two

Random Integers

Word Lengths

Average Length 1 10x = 2 10x =

d. Repeat the process one more time, this time choosing 25 words. Calculate the average

length of sample of 25 words and record as 25x . Also record the mean length on a

post-it note. Make sure the post-it is labeled!

Random Integers

Word Lengths

Average Length 25x =

Clear a space on the floor, the wall, or the board. Use masking tape to make a number line

(horizontal axis) with average lengths marked from 2 to about 6 on the axis, with tick marks

every 0.2. Each interval should be a little more than the width of a post-it note.

e. Place the post-its with the average length of the samples of 5 words on the axis according

to the average length, making a post-it dotplot on the floor/wall/board.

Look at the shape of the final dotplot. Describe the distribution of the words’ lengths.





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Make a second axis on the floor and label it with increments of 0.1. Again, each interval should

be a little more than the width of a post-it note.

f. Plot the means for the sample size of 10. What is the shape of the dotplot for the

distribution of 10x ? How does it compare with the previous distribution?

Make a third axis on which to create a dotplot of the means for the sample size 25.

g. Plot the means for the sample size of 25. What is the shape of the dotplot for the

distribution of 25x ? How does it compare with the previous distribution?

h. Divide into groups for the next part of this task. There should be at least 4 groups. Each

group will take one of the dotplots above, as well as the entire list of words, and find the

mean and standard deviation of the sample means. (Because there are so many words,

two groups should separately determine the overall mean and standard deviation and

check each other.)

Post a table similar to the one below on the board so that groups can post their results.

Record all groups’ result here.

Mean Standard Deviation Shape of the Distribution

Population

Samples of 5

Samples of 10

Samples of 25

i. Previously, we stated that if samples were taken from a normal distribution, that the mean

and standard deviation of the sampling distribution of sample means was also normal

with x and xn

. In this activity, we did not begin with a normal distribution.

However, compare the means for the samples of 5, 10, and 25 with the overall mean of

the word lengths. Then compare the standard deviations with the standard deviation of all

the word lengths. Do these formulas appear to hold despite the population of word

lengths being obviously non-normal? Explain.

j. This brings us to the Central Limit Theorem (CLT) for Sample Means

Choose a simple random sample of size n from any population, regardless of the original

shape of the distribution, with mean and finite standard deviation . When n is large,

the sampling distribution of the sample mean x is approximately normal with mean

______ and standard deviation ________.





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Note: The statement “when n is large” seems a bit ambiguous. A good rule of thumb is

that the sample size should be at least 30, as we can see in the dotplots above. When the

sample sizes were sample, i.e. 5 and 10, the plots were still quite right skewed.

3. Applying the CLT for Sample Means

Whenever we have a normal distribution, we are able to use normal tables and normal

calculations to find probabilities. Due to the CLT, we can use normal calculations to determine

probabilities about sample means drawn from large samples.

a. First, we need to recall how to standardize scores, that is, find z-scores. State the formula

for z-scores that you learned in Math 3.

Recall that whenever we standardize values, we take the estimate (or statistic) minus the

corresponding parameter and divide that difference by the corresponding standard

deviation. We will standardize sample means in the same way. Substitute the statistics

and parameters for sample means into the z-score formula to obtain our standardization

formula for sample means.

b. Consider an IQ test with scores that vary according to a normal distribution with a mean

of 100 and a standard deviation of 15. Find the probability that a single person scores

higher than 110.

c. Suppose you take a sample of 20 individuals who took this IQ test. What are the mean

and standard deviation of the sampling distribution of the average score of this size

sample?

d. Find the probability that the mean score of these 20 people is greater than 110.

e. Find the probability that the mean score of these 20 people is between 95 and 110.

f. How would you expect your answers to parts d and e would be different if the sample

size were 50 instead of 20?

g. Calculate these probabilities and comment on your conjecture.

h. Which of your answers, if any, to parts b, c, d, e, and g would be affected if the

distribution of the candy bar weights was not normally distributed? Explain.





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The Gettysburg Address

Four score and seven years ago our fathers brought forth on this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all men are created equal. Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived and so dedicated, can long endure. We are met on a great battle-field of that war. We have come to dedicate a portion of that field, as a final resting place for those who here gave their lives that that nation might live. It is altogether fitting and proper that we should do this. But, in a larger sense, we cannot dedicate -- we can not consecrate -- we can not hallow -- this ground. The brave men, living and dead, who struggled here, have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us -- that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion -- that we here highly resolve that these dead shall not have died in vain -- that this nation, under God, shall have a new birth of freedom -- and that government of the people, by the people, for the people, shall not perish from the earth.





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Gettysburg Address Word List (page 1)

Numbe

r

Word Lengt

h

Numbe

r

Word Leng

th

Numbe

r

Word Lengt

h

001 Four 4 046 nation 6 091 live. 4

002 score 5 047 so 2 092 It 2

003 and 3 048 conceived 9 093 is 2

004 seven 5 049 and 3 094 altogether 10

005 years 5 050 So 2 095 fitting 7

006 ago. 3 051 dedicated, 9 096 and 3

007 our 3 052 Can 3 097 proper 6

008 fathers 7 053 Long 4 098 that 4

009 brought 7 054 endure. 5 099 we 2

010 forth 5 055 We 2 100 should 6

011 upon 4 056 Are 3 101 do 2

012 this 4 057 met 3 102 this. 4

013 continent 9 058 on 2 103 But 3

014 a 1 059 A 1 104 in 2

015 new 3 060 great 5 105 a 1

016 nation: 6 061 battlefield 11 106 larger 6

017 conceived 9 062 of 2 107 sense, 5

018 in 2 063 That 4 108 we 2

019 Liberty, 7 064 war. 3 109 cannot 6

020 and 3 065 We 2 110 dedicate, 8

021 dedicated 9 066 Have 4 111 we 2

022 to 2 067 Come 4 112 cannot 6

023 the 3 068 To 2 113 consecrate, 10

024 propositio

n

11 069 dedicate 8 114 we 2

025 that 4 070 a 1 115 cannot 6

026 all 3 071 portion 7 116 hallow 6

027 men 3 072 Of 2 117 this 4

028 are 3 073 That 4 118 ground. 6

029 created 7 074 field 5 119 The 3

030 equal. 5 075 as 2 120 brave 5

031 Now 3 076 A 1 121 men, 3

032 we 2 077 final 5 122 living 6

033 are 3 078 resting 7 123 and 3

034 engaged 7 079 place 5 124 dead, 4

035 in 2 080 For 3 125 who 3

036 A 1 081 those 5 126 struggled 9

037 great 5 082 Who 3 127 here 4

038 civil 5 083 Here 4 128 have 4

039 war, 3 084 Gave 4 129 consecrated 11

040 testing 7 085 their 5 130 it, 2

041 whether 7 086 lives 5 131 far 3

042 that 4 087 That 4 132 above 5

043 nation, 6 088 That 4 133 our 3

044 or 2 089 nation 6 134 poor 4

045 any 3 090 might 5 135 power 5





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Gettysburg Address Word List (page 2)

Number Word Length Number Word Length Number Word Length

136 to 2 181 Have 4 226 we 2

137 add 3 182 Thus 4 227 here 4

138 or 2 183 Far 3 228 highly 6

139 detract. 7 184 So 2 229 resolve 7

140 The 3 185 Nobly 5 230 that 4

141 world 5 186 advanced. 8 231 these 5

142 will 4 187 It 2 232 dead 4

143 little 6 188 Is 2 233 shall 5

144 note, 4 189 rather 6 234 not 3

145 nor 3 190 For 3 235 have 4

146 long 4 191 Us 2 236 died 4

147 remember 8 192 Here 4 237 in 2

148 what 4 193 to 2 238 vain, 4

149 we 2 194 Be 2 239 that 4

150 say 3 195 dedicated 9 240 this 4

151 here, 4 196 To 2 241 nation, 6

152 but 3 197 The 3 242 under 5

153 it 2 198 Great 5 243 God, 3

154 can 3 199 Task 4 244 shall 5

155 never 5 200 remaining 9 245 have 4

156 forget 6 201 before 6 246 a 1

157 what 4 202 us, 2 247 new 3

158 they 4 203 That 4 248 birth 5

159 did 3 204 From 4 249 of 2

160 here. 4 205 These 5 250 freedom, 7

161 It 2 206 honored 7 251 and 3

162 is 2 207 Dead 4 252 that 4

163 for 3 208 We 2 253 government 10

164 us 2 209 Take 4 254 of 2

165 the 3 210 increased 9 255 the 3

166 living, 6 211 devotion 8 256 people, 6

167 rather, 6 212 to 2 257 by 2

168 to 2 213 That 4 258 the 3

169 be 2 214 Cause 5 259 people, 6

170 dedicated 9 215 To 2 260 for 3

171 here 4 216 Which 5 261 the 3

172 to 2 217 They 4 262 people, 6

173 the 3 218 Gave 4 263 shall 5

174 unfinished 10 219 The 3 264 not 3

175 work 4 220 Last 4 265 perish 6

176 which 5 221 Full 4 266 from 4

177 they 4 222 measure 7 267 the 3

178 who 3 223 Of 2 268 earth. 5

179 fought 6 224 devotion, 8

180 here 4 225 That 4





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CONFIDENCE INTERVALS LEARNING TASKS

Statistical Inference provides methods for drawing conclusions about a population from sample

data. The two major types of statistical inference are confidence intervals and tests of

significance. We will focus only on understanding and using confidence intervals in Math 4.

Tests of significance are a major focus of Statistics courses.

1. The Empirical Rule

a. Suppose that the mean SATM score for seniors in Georgia was 550 with a standard

deviation of 50 points. Consider a simple random sample of 100 Georgia seniors who

take the SAT. Describe the distribution of the sample mean scores.

b. What are the mean and standard deviation of this sampling distribution?

c. Use the Empirical Rule to determine between what two scores 68% of the data falls, 95%

of the data falls, and 99.7% of the data falls.

For the 95% interval, this means that in 95% of all samples of 100 students from this population,

the mean score for the sample will fall within ___ standard deviations of the true population

mean or ____ points from the mean.

2. Confidence Intervals

In the above problem, we took the mean and added/subtracted a certain number of standard

deviations. That is, we calculated 2 2x x xn

for the 95% interval and

3 3x x xn

for the 99.7% interval.

The interval of numbers found, i.e. (540, 560) is called a 95% confidence interval for the

population mean.

Above, we knew the population mean, but in practice, we often do not. So we take samples and

create confidence intervals as a method of estimating the true value of the parameter. When we

find a 95% confidence interval, we believe with 95% confidence that the true parameter falls

within our interval. However, we must accept that 5% of all samples will give intervals which do

not include the parameter.





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Every confidence interval takes the same shape: estimate margin of error. In the interval (540,

560), the margin of error is 10.

The margin of error has two main components: the number of standard deviations from the mean

(i.e. the z-score) and the standard deviation. (Margin of error = z .)

Because we do not usually know the details of a population parameter (e.g. mean and standard

deviation), we must use estimates of these values. So our margin of error becomes m =

z( estimate). Therefore, the confidence interval becomes

estimate margin of error estimate z( estimate).

The z-score used in the confidence interval depends on how confident one wants to be. There are

a few common levels of confidence used in practice: 90%, 95%, and 99%.

The Empirical Rule provides estimates for the amount of data within specified numbers of

standard deviations, and therefore, can help us find approximate intervals for being 68%, 95%,

and 99.7% confident that we have included the true population parameter. Let’s find closer

estimates for the number of standard deviations from the mean within which certain percentages

of data lie.

a. Within how many standard deviations of the mean would one locate the middle 95% of

the data? (Hint: Draw a picture and use the normal table or invNorm on your calculator.)

b. Within how many standard deviations of the mean would one locate the middle 90% of

the data? (Hint: Draw a picture and use the normal table or invNorm on your calculator.)

c. For the common confidence levels, then, we have the following z-scores, called z*.

Complete the following table using your answers above.

Confidence

level

z*

90%

95%

99% 2.576

For any confidence intervals you are expected to compute by hand in Math 4, you will use these

z* values. Thus, our final form of the confidence interval is estimate z*( estimate).

You will continue investigating confidence intervals and, specifically, margin of error, through

the next two activities.





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3. President’s Approval Ratings: Part 1 (Confidence Intervals for Proportions)

a. In the article at the beginning of this unit, Bush’s approval rating was 41 percent with a

margin of error of plus or minus 3 points. Write this statement as a confidence interval.

b. The article does not state the confidence level for the ratings. Use the margin of error to

determine the standard error of the estimate (standard deviation) if a 95% confidence level

was used. What is the standard error if a 99% confidence level was used?

c. What is the formula for calculating the standard deviation of sample proportions? In

practice, we do not know the true population parameter. Instead, we must estimate using

the sample proportion. Rewrite this equation using the symbol for the sample proportion.

This is called the standard error of the sample proportion.

p̂ SE =

d. If the Gallup poll in the article used a 95% confidence level, what was the sample size?

What if the poll used a 99% confidence level? (Use the answers to b and c. Recall that the

sample proportion was 41%.) Refer to the article. How many people were polled? What

does that tell you about the confidence level employed?

e. Assume the pollsters were undecided about how many citizens to include in their poll.

However, they knew that they wanted the margin of error to be 3 percent or less and they

wanted to be 95% confident in their results.

That is, they wanted ˆ ˆ1

*p p

z mn

or ˆ ˆ1

1.96 .03p p

n.

What value of p will always give the greatest margin of error? Make a conjecture. Explain

your reasoning. Compare with a neighbor. Call this value p*.

f. Increase the sample sizes you found in part e by 500. Still using your p*, determine the

margin of error in both the 95% and 99% cases for intervals with these new sample sizes.

How different is the margin of error? (Margin of error = z*SEestimate.) What do you

notice?

g. Decrease the sample sizes you found in part e by 500. Still using your p*, determine the

margin of error in both the 95% and 99% cases for intervals with these new sample sizes.

How different is the margin of error?





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h. Finally, still using your p*, determine the margin of error in both the 95% and 99% cases

for intervals with sample sizes of 200 people. How different is the margin of error?

(Margin of error = z*SEestimate.) What do you notice?

i. Based on your answers to f , g, and h, how does the margin of error behave when the

sample size changes? Use a graph of 1

yx

to support your answer. (How does

1y

x relate to our problem?)

j. Look back through these examples, how does the confidence level affect the margin of

error? Why does this make sense in the context of confidence intervals?

4. President’s Approval Ratings: Part 2 (Confidence Intervals for Proportions)

As we saw in the last problem, the form of a confidence interval for the population proportion p

is estimate z*( estimate) or ˆ ˆ1

ˆ *p p

p zn

.

a. Suppose you are planning to conduct a presidential approval survey in your school. You are

planning to construct a 95% confidence interval of the proportion of students who approve

of the job the president is doing and you want a margin of error of no more than 5 points.

Determine the minimum number of people you need to survey.

b. Suppose you surveyed 50 students and 34 said that they approve of how the president is

doing. Construct 90%, 95%, and 99% confidence intervals for the true proportion of students

in your school who approve of how the president is doing. Be sure to specifically calculate

the margin of error.

c. Explain, in context, what it means to be 95% confident. Are you sure that the true proportion

is within this interval? Explain.

d. Based on the results above, write a sentence or two about how the students at the school

view how well the president is doing his job. Back up any conclusions.

e. We can also use a calculator to construct confidence intervals.

On your TI calculator, use the following keystrokes: STAT – TESTS – 1-PropZInt. What do you

think 1-PropZInt means?

Using the information from b, find the 95% confidence interval. How does this interval compare

with the one you found previously?





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f. Suppose you know that a sample of 350 students was collected and that 58% gave the

president a positive approval rating. Explain how to use the calculator to find a 90%

confidence interval for the true proportion of students at the school who would give a

positive approval rating. Find the confidence interval.

g. The calculator makes finding confidence intervals quite easy; however, you do not always

obtain all the information you want from the calculator output. For example, you are not

given the margin of error in the output. How can you use the output to determine the margin

of error? Illustrate, using an 88% confidence interval for the survey in this problem (x = 34,

n = 50).

h. Let’s do a simulation. Suppose that 55% of the students in your school have a positive

opinion of the president. We are going to generate a sample of 50 of those students. Input the

following into your calculator: randBin(1, .55, 50)L1:sum(L1). Why is it appropriate to

simulate this situation with a binomial distribution?

The 1 in the formula indicates a success. The 55% is the probability of success. 50 is the

sample size. The values will be stored in List 1, but the calculator will also sum the values in

List 1, returning only this sum. What does the sum represent?

i. Now, construct a 95% confidence interval for the true proportion of positive approval

ratings. Does 55% fall in your interval? Did you expect this? Explain.

j. Repeat simulating and constructing confidence intervals four more times. List your

confidence intervals below.

k. As a class, make a tally of how many confidence intervals contained 55% and how many did

not. Are you surprised at the result? Explain.

l. The margin of error in the above confidence intervals is quite large. Name two ways we

could reduce the margin of error.

5. Racing Hearts (Confidence Intervals for Means)

For this next activity, we will turn our attention to finding confidence intervals for population

means.

a. State the estimate and standard deviation for estimating population means from sample

means.

b. Recall that the form of a confidence interval is estimate z*( estimate). Write the statement

of the confidence intervals for means by substituting the values from above.





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c. Because we generally do not know the true mean and standard deviation, we must

estimate them. What is the estimate for ? For the standard deviation, instead of , we use

the sample standard deviation, s. Make the appropriate substitution in the confidence

interval from above.

d. When we do not know the population standard deviation and instead use s

n, the

distribution is no longer exactly a normal distribution. Instead, it takes on a t-distribution.

The t-distribution is similar to the normal. To see this, graph the following. (The

commands are under the DISTR menu.)

Set your window to be the following: X[-5, 5] and Y[0, .5].

Y1 = normalpdf(X) Y2 = tpdf(X, 2) Y3 = tpdf(X,30)

Y2 would be the t-distribution for a small sample size, whereas Y3 is the t-distribution for

a sample size of 31.

What do you notice about the graphs of Y1, Y2, and Y3?

For the purpose of this course, we need to recognize that we are using a t-distribution.

This alters our confidence interval slightly to be *s

x tn

. We will use technology to

calculate these confidence intervals.

e. Let’s collect some data! We want to estimate the average resting heart rate of a typical

senior. Take your pulse for a full 60 seconds. Write it down. Form a group of five students

and record each of their pulses. Input the data into List 1 of your calculator.

f. Using the calculator, construct a 95% confidence interval for the average resting heart

rate. To do this, use the following key strokes: STAT – TESTS – Tinterval. Because we

have data, choose Data. Make sure that List 1 is indicated, with a frequency of 1, and a C-

level of 95. Record the mean, standard deviation, the confidence interval.

g. Now, combine with another group of 5. Repeat part f with all 10 sets of data.

h. As a class, collect at least 30 resting heart rates. Enter into L1 and calculate the confidence

interval. Write a sentence or two reporting what you believe to be true about the average

resting heart rate of seniors in your school.

i. Suppose you were planning a larger study of heart rates. Based on these results, you

estimated the standard deviation to be 4.9. You would like to form a 99% confidence





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interval and have a margin of error no more than 3 beats from the true mean. How many

students must you include in your study? (Use z* instead of t*.)

j. Suppose you did collect this data from 200 students. The average resting heart rate was

58.6 beats per minute and the standard deviation was 4.8. Use the calculator to find a 99%

confidence interval for the mean resting heart rate of students at your school. Interpret

your results in the context of the problem. (Note: For this problem, we have Stats instead

of Data. Make the appropriate change in the TInterval screen.)

6. Synthesis of Confidence Intervals

Write a brief review sheet or notes that you would give to an absent classmate outlining the

major ideas about confidence intervals and how to construct them.