AP Statistics Name: ____________________________ Chapter 2 – Modeling Distributions of Data Date: ____________________ Per: _____

Section 2.2 Part 1: Density Curves and Normal Distributions pgs. 103 - 112Today, you will learn how to:o Use a density curve to model distributions of quantitative data.o Identify the relative locations of the mean and median of a distribution from a density curve.o Use the 68-95-99.7 Rule to estimate the proportion of values in a specified interval, or to estimate the value that

corresponds to a given percentile in a Normal distribution.

Activity 1: Exploring Density Curves (3 Labs)

Experiment 1: Roll a die and record the value it lands on. 1st roll: __ 2nd roll: __ 3rd roll: ___ 4th roll: ___ 5th roll: ___

Prediction: Actual:

Experiment 2: Try to toss a penny and make it land on the target. Measure the distance of the penny from the target in inches. Round to the nearest tenth.

1st Attempt: _______ 2nd Attempt: ________ 3rd Attempt: _______ 4th Attempt: ________ 5th Attempt: ______

Prediction: Actual:

Experiment 3: Try to stop your stopwatch at exactly 5 seconds. Record what the stopwatch reads below. Record to the tenths place.

1st Attempt: _______ 2nd Attempt: ________ 3rd Attempt: _______ 4th Attempt: ________ 5th Attempt: ______

Prediction: Actual:

Normal Curves: Label the values 1, 2, and 3 standard deviations above and below the mean using the stopwatch data.

What percentage of the data is within two standard deviations of the mean?

What percentage of the data is further than two standard deviations from the mean?

What percentage of the data is greater than 1 standard deviation above the mean?

What percentage of the data is between z = -1 and z = 2?

Big Ideas from Section 2.1 Part 1: Describing Location in a Distribution

Density Curves Types of curves and mean/median:

The Normal Density Curve

Check Your Understanding (Groupwork):

1. An Internet reaction time test asks subjects to click their mouse button as soon as a light flashes on the screen. The light is programmed to go on at a randomly selected time after the subject clicks “Start.” The density curve models the amount of time the subject has to wait for the light to flash.

a. What height must the density curve have? Justify your answer.

b. About what percent of the time will the light flash more than 3.75 seconds after the subject clicks “Start”?

c. Calculate and interpret the 38th percentile of this distribution.

2. Use the figure to answer the following questions.

a. Explain why this is a legitimate density curve.

b. About what proportion of observations lie between 7 and 8?

c. Mark the approximate locations of the median and the mean on the curve. Explain why the mean and the median have the relationship that they do in this case.

3. The distribution of heights of young women aged 18 to 24 is approximately Normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.

a. Sketch the Normal curve that approximates the distribution of young women’s height. Label the mean and the points that are 1, 2, and 3 standard deviations from the mean.

b. About what percent of young women have heights less than 69.5 inches? Show your work.

c. Is a young woman with a height of 62 inches unusually short? Justify your answer.

4. On a standardized test, Maggie scored 86, exactly one standard deviation above the mean. If the standard deviation for the test is 6, what is the mean score for the test?

5. The heights for girls are normally distributed with a mean of 66 inches. If 95% of the heights of these girls are between 63 and 69 inches, what is the standard deviation for this group?

6. A set of scores with a normal distribution has a mean of 50 and a standard deviation of 7. Approximately what percent of the scores fall in the range 36-64?

7. The mean of a normally distributed set of data is 52 and the standard deviation is 4. Approximately 95% of all the cases will lie between which measures?

8. Battery lifetime is normally distributed for large samples. The mean lifetime is 500 days and the standard deviation is 61 days. Approximately what percent of batteries have lifetimes longer than 561 days?

9. A test was given to 120 students and the scores approximated a normal distribution. If the mean was 72 and the standard deviation was 9, about what number of students scored an A- or higher (above 90%)?

10. Hair lengths have a mean of 9 inches and a standard deviation of 3. Approximately what percent of hair lengths fall within the range 6 to 12 inches? (This a trick question).

AP Statistics Name: ____________________________ Chapter 2 – Modeling Distributions of Data Date: ____________________ Per: _____

Section 2.2 Part 2: Density Curves and Normal Distributions pgs. 113 - 117Today, you will learn how to:o Use Table A and technology to find the proportion of values in a specified interval.o Use Table A and technology to estimate the value that corresponds to a given percentile in a Normal distribution.

Activity 1: Back to the Future After accelerating for 20 seconds, a DeLorean sports car has a wide range of speeds that it can achieve, depending on traction. The distribution of speed follows an approximately Normal distribution with a mean of 80 mph and a standard deviation of 7.7 mph.

1. Label the appropriate values, z-scores, and proper notation on the normal distribution

2. What percentage of the runs will give the Delorean a speed greater than 87.7 mph?

3. What percentage of the runs will give the Delorean a speed between 64.6 mph and 87.7 mph?

4. What percentage of the runs will give the Delorean a speed less than 64.6 mph?

5. What percentage of the runs will give the Delorean a speed less than 68.45 mph?

6. What percentage of the runs will give the Delorean a speed greater than 85 mph? Show work & pictures.

7. What percentage of the runs will give the Delorean a speed between 70 and 95 mph? Show work & pictures.

8. Marty wants his last run to be in the top 15% of all the possible speeds. What speed does he need to achieve to be in the top 15%? Show work and pictures.

Big Ideas from Section 2.1 Part 2: Describing Location in a Distribution

Check Your Understanding

1. Use Table A to draw and then find the area to the left of (less than): (Shade it in !!!)

(a) z= 0.81: _______________________ (b) z= 0.45: _____________________

(c) z = −¿2.27________________________ (d) z= −¿ 0.45: _____________________

2. Use Table A to draw and then find area to the right of (more than): (Shade it in !!!)

(a) z= 1.92: _______________________ (b) z= 0.63: _____________________

(c) z = −¿ 1.27________________________ (d) z= −¿ 0.17: _____________________

3. Suppose we wanted to find the proportion of observations in a Normal distribution that were more than 1.53 standard deviations above the mean. That is, find the proportion of observations in the standard

Normal distribution are greater than z = 1.53, or in other words, find the area for z > 1.53. Draw a picture and show your work for full credit.

4. Find the proportion of observations from the standard Normal Distribution if -1.25 < z < 0.81. Draw a picture and show your work for full credit.

5. Find the proportion of observations from the standard Normal distribution if −0.58 < z < 1.79. Draw a picture and show your work for full credit.

6. Working Backwards: Find the z-score with the 90th percentile of the Normal Standard Curve. Draw a picture and show your work for full credit.

7. In a standard Normal distribution, 7.49% of the observations are below what z value? Draw a picture

and show your work for full credit.

8. When professional golfer Jordan Spieth hits his driver, the distance the ball travels can be modeled by a Normal distribution with mean 304 yards and standard deviation of 8 yards.

a) On a specific hole, Jordan would need to hit the ball at least 290 yards to have a clear second shot that avoids a large group of trees. What percent of Spieth’s drives travel at least 290 yards? Draw a picture and show your work for full credit.

b) On another golf hole, Spieth has the opportunity to drive the ball onto the green if he hits the ball a distance in the top 10% of all his drives. How far does the ball have to go? Draw a picture and show your work for full credit.

AP Statistics Name: ____________________________ Chapter 2 – Modeling Distributions of Data Date: ____________________ Per: _____

Section 2.2 Part 3: Normal Distribution Calculations (CRITICAL!!!) pgs. 117 - 121Today, you will learn how to:o Write the Three-Step Process for using Table A and/or technology to find the proportion of values in a specified

interval, or to estimate the value that corresponds to a given percentile in a Normal distribution.

PLEASE READ BELOW: The Three Step Process for Normal Distribution Calculations

Here is an outline of the 3-STEP PROCESS for finding the proportion of a Normal distribution in any region. YOU WILL BE EXPECTED TO FOLLOW THIS PROCESS PERFECTLY. For those of you that did not understand what I just said, read slowly... DO. NOT. DEVIATE. FROM. THE. 3 STEPS. OR. TAKE. ANY. SHORTCUTS. Unless you want to get zero credit. SO Follow the sentence frames to a T! Or pay the price…

Sentence Frames:Step 1: The ___________________________________ follows a Normal distribution with ________ and _________. We want to find _________________________, as indicated on the Normal Curve drawn to the ___________ (DRAW IT!! LABEL IT!! TITLE IT!!)Step 2: The standardized score for the boundary value(s) is/are

Z= X−μσ = show work = answer. According to Table A, the

proportion of values above/below/in-between ___ is ________. Step 3: Rephrase the original question into an answer in context with the proportion found in step 2.

EXAMPLE #1: Rafael Nadal’s Serving Speed (Tennis ladies, this one’s for you)

In a past tournament, tennis player Rafael Nadal averaged 115 miles per hour (mph) on his serves. Assume that the distribution of his serve speeds is Normal with a standard deviation of 6 mph.

(a) About what percent of Nadal’s serves would you expect to exceed 120 mph? Show your method.

(b) What percent of Nadal’s serves are between 100 and 110mph? Show your method.

EXAMPLE #2: Using Table A in Reverse (see example on page 120)

According to the CDC’s Growth Charts, the heights of 3-year old females are approximately Normally distributed with a mean of 94.5 cm and a standard deviation of 4 cm. What is the third quartile of this distribution?

FOLLOW THE 3-STEP PROCESS ON HW#7 AND HW#8 IF YOU WANT TO BE ABLE TO ACE THESE QUESTIONS ON EXAM #1 NEXT WEEK! You will produce what you practice over the week. As Vince Lombardi wisely said,

“Practice does not make perfect. Only perfect practice makes perfect.”There are no shortcuts to desirable results when it comes to your knowledge. So get practicing!

AP Statistics Name: ____________________________ Chapter 2 – Modeling Distributions of Data Date: ____________________ Per: _____

Section 2.2 Part 4: Assessing Normality pgs. 122 - 127Today, you will learn how to:o Write the Three-Step Process for using Table A and/or technology to find the proportion of values in a specified

interval, or to estimate the value that corresponds to a given percentile in a Normal distribution.

Opening Question: What do we know about “Normal” distributions in terms of their characteristics?

EXAMPLE : No Space in the Fridge?

The measurements listed below describe the usable capacity (in cubic feet) of a sample of 36 side-by-side refrigerators (Consumer Reports, May 2010).

12.9 13.7 14.1 14.2 14.5 14.5 14.6 14.7 15.1 15.2 15.3 15.315.3 15.3 15.5 15.6 15.6 15.8 16.0 16.0 16.2 16.2 16.3 16.416.5 16.6 16.6 16.6 16.8 17.0 17.0 17.2 17.4 17.4 17.9 18.4

(a) Are the data close to Normal?

Step 1: Plot the data: Step 2: Determine the summary statistics:

Mean: x= _________ Median: _________

Standard Deviation: Sx=¿¿

Step 3: count the number of observations within one, two, and three standard deviations of x.

Mean± 1Sx :¿ Percent of observations: ______________________

Mean± 2 Sx :¿ Percent of observations: ______________________

Mean± 3 Sx :¿ Percent of observations: _______________________

What do you conclude? Are the data close to Normal?

(b) Here is a Normal probability plot of the refrigerator data from part (a). Does it support your answer from part (a)? Why or why not?

Big Ideas from Section 2.1 Part 4: Assessing Normality

5

10

15

20

25

Area (thousands)0 200 400 600 800

-2

-1

0

1

2

Area (thousands)0 200 400 600

Normal Quantile = -0.79 + 1.04e-05Area

st_size Normal Quantile Plot

Now You Try: State Areas

1. Use the histogram and Normal probability plot below to determine if the distribution of areas for the 50 states is approximately Normal.

2. Here are the survival times in days of 72 guinea pigs after they were injected with infectious bacteria in a medical experiment. Survival times, whether of machines under stress or cancer patients after treatment, usually have distributions that are skewed to the right. Determine whether these data are approximately Normally Distributed.

Section 2-2 Review Worksheet

For each question, be sure to draw the appropriate pictures and show the appropriate formulas. Show your work as if this was a quiz or a test.

Mr. Wilcox spends a full day at Gingerman Raceway testing some new tires on his Honda Civic. For 400 laps, he records his laptime in minutes. At the end of the day, he plots the distribution of times and realizes that it follows an approximately normal distribution with a mean of 1.84 and a standard deviation of .07.

1. Mr. Wilcox would really like his lap times to be consistently between 1.75 and 1.85 minutes. What proportion of his laps were in this range? Interpret your result in the context of the problem.

2. In order to beat ItaTom, Mr. Wilcox needs to have lap times that are less than 1.72 minutes. Find the proportion of laps that were faster than 1.72 minutes. Interpret your result in the context of the problem.

3. In what proportion of laptimes did it take Mr. Wilcox more than 2 minutes? Interpret your result in the context of the problem.

4. What laptime is necessary for Mr. Wilcox to be in the fastest 3% of all his laps? Interpret your result in the context of the problem.