lesson 1.1.1 the statistical analysis process 1/students/pdf/lesson_1... · “is astrology real,...

© 2011 THE CARNEGIE FOUNDATION FOR THE ADVANCEMENT OF TEACHING A PATHWAY THROUGH STATISTICS, VERSION 1.5, STATWAY™ - STUDENT HANDOUT

STATWAY™ STUDENT HANDOUT

Lesson 1.1.1 The Statistical Analysis Process

STUDENT NAME DATE

INTRODUCTION What is statistics? Why do we study statistics? Statistics is about using data (or numeric observations about the world) to answer questions. Before

scientists created statistics and before people used data, they would make arguments about how the world

worked based on opinions and hunches. A lot of times these arguments were wrong. For example,

hundreds of years ago, people argued the earth was the center of the universe. Looking at data allowed us

to start making better conclusions. Through science, we saw that the earth goes around the sun, not the

other way around. But science and mathematics weren’t enough, because sometimes data were really

messy. Things often don’t happen the same way every time. And sometimes what looks like a trend is

actually due to random chance. So to understand the data in the world, scientists developed statistics. Using

statistics they could know whether something that seemed to happen some of the time was actually true, or

if it was just random.

There are lots of things you might want to know about the world--things that statistics can help you answer.

It could help you in your job. As a faculty member i may want to know whether a new teaching method

works better for my students. Or if you worked for a medicine company, you may want to know whether a

new medicine can help people who have cancer. Statistics can also help you understand society so we can

have a better democracy. Politicians and news organizations conduct surveys and polls in order to find out

what voters are concerned about and who might win an election. Statistics can help you know whether the

results of surveys and polls tell you something new, or whether the news story is overblown.

In addition, when you buy things statistics can help. You probably hear news stories about how oatmeal or

red wine are good for your health and hydrogenated oils are bad for you. You might wonder: how do

researchers know this? And how certain are they? Certainly not every person who drinks red wine lives to

old age, and not every person who eats those oils dies immediately. With all the complexity in the real

world, how do we know whether the relationship between some food and your health is real or imaginary?

Statistics can help you do just that.

In general, in statistics, we formulate a question that we want answered--like about the impact of a

STATWAY STUDENT NOTES | 2



medicine or the health effects of a food. Then we don’t just argue about it in theory. We gather data from

the real world and then we summarize the data so we can see if the result we’re expecting is there, or if it

isn’t. In addition, statistics gives us the tools to know how confident we are that our answer is correct.

Today we are going to think about this with a question that lots of people wonder but that we hardly ever

use real data to answer. “Is astrology real, or is it just random?” Signs of the Zodiac and Personality Traits The table below lists the four steps in this process.

Steps in a Statistical Investigation

1. Ask a question that can be answered by collecting data.

2. Decide what to measure and then collect data.

3. Summarize and analyze the data.

4. Draw a conclusion and communicate the results.

Can a person’s birthday determine his or her personality traits, like being kind or being jealous? In this task,

you will try to figure out whether personality characteristics are determined by a person’s birthday. Your

instructor just gave you a list of personality traits. These traits are in birthday groups. Each birthday group

has 3 sets of personality traits. Only 1 set actually goes with that birthday group according to the zodiac

calendar. The other 2 sets are not associated with the zodiac or astrological sign for those birthdays.

Look for your birthday on the handout. Read the 3 sets of personality traits for your birthday group. Pick

the set of personality traits that you think matches your own personality. Of course, none of the sets will

match your personality perfectly, so just pick the set that describes your personality the best.

Below write the number of the set you picked.




TRY THESE 1 Some people are always jokesters. Some people may great attention to details. These are some of

many personality traits that people have. Do you think your birth date determines your

personality traits?

How can we use the choices made by everyone in the class to answer this question: does a

person’s zodiac sign play a part in determining his or her personality traits? Write your answer

below.

2 Personality may or may not be related to astrological sign. If the theory that personality traits are

related to astrological sign is not correct, will some students still pick the correct set of personality

traits? Tell why you think this.

About what fraction of the students in the class do you expect to pick the description that astrologists

say matched their sign? Tell why do you think this?

3 Imagine the astrologists’ theory that the zodiac sign determines personality is correct. What

would be true about the fraction of students who pick the correct set of personality traits? Tell

why you think this.




Think about your entire class. How large of a fraction of your class would need to pick the correct

description for you to be convinced that the theory might be true? Tell why you think this.

4 Imagine that half the students in your class select the correct set of traits. Does this guarantee

that the theory could be true? If not, give another explanation for why so many students picked

the correct set of traits.

NEXT STEPS Ruling Out Chance In the last activity, you learned:

When the zodiac sign has nothing to do with personality traits, you expect the fraction of students in the

class who pick the list of personality tracks that corresponds to their zodiac sign to be around 1/3.

5 Your instructor has given you three cards, one with “Match” and two with “No Match” written on

them. Mix the cards and choose one of them. Complete the following information.

My card says:

Number of students in the class:

Number of students who selected the card that said “Match”:

Fraction of the students who selected “Match”:




Proportion of the students who selected “Match”:

Helpful hint: A proportion is a decimal between 0 and 1. You can calculate a proportion by dividing the

numerator of the fraction by the denominator. For example, if the fraction of students who picked “match”

is 5/7, then you divide 5 by 7. The proportion would be 0.714.

6 If your class repeated this process a second time, would you get exactly the same “Match”

proportion for the class? Why or why not?

7 If your class repeated this process a second time, would you get exactly the same “Match”

proportion for the class? Why or why not?

8 Each of your classmates mixed the cards and picked one at random. You then computed the

proportion of students in the class that picked the “Match” card. Tell why knowing this one

proportion is not enough to determine the values of the “Match” proportion you expect to

observe just by chance.

9 You know that knowing this proportion is not enough. What additional information would help

you get a better understanding of the proportion of matches you would expect to observe just by

chance?




10 You can repeat the process of getting “by chance” outcomes a large number of times. This will

help you understand what kinds of match proportions are consistent with picking cards at

random.

First, write the class proportion from Task 2 in the following table under Trial 1. Second, work

with your classmates to repeat the process of picking cards until you have proportions for 10

trials.

When you are done, fill in the table below with the proportions of the students who picked “match.” (Note: “Observed Proportion” means the proportion we see from the data we collected in each trial.)

Trial Observed Proportion

1

2

3

4

5

6

7

8

9

10

11 Your professor constructed a dotplot which shows the observed “Match” proportions for the 10

trials where the students picked one of three choices at random.

A What was the smallest match proportion observed? B What was the largest match proportion observed?




C Did the match proportion differ much from trial to trial?

12 Summarize what your dotplot tells you. Why did you create this graph?

13 We can compare Task 2 with Task 1 about the relationship between the zodiac sign and

personality traits. We can do this because you picked 1 set out of 3 possible sets of traits. You did

this to test the zodiac sign theory that there is a relationship between your personality traits and

your zodiac sign. Then, in task 2, you picked 1 card out of 3 possible cards. They will have similar

proportions--1 out of 3 chance of getting a matching set of traits and picking the “match” card.

Now, look at the dotplot. How does this graph help tell you whether the actual class data from

Task 1 support the zodiac sign theory?

14 Use the dotplot to answer the following question?

If students in the class were picking at random, it would be unusual to see a match proportion as

large as ______________.

15 Suppose that the proportion of students who picked the set of personality traits that matched

their zodiac sign was 0.40. Does this tell you that personality traits are unrelated to sign? Explain

your answer using the dotplot.




16 How large does the proportion of students who picked matching sets have to be to convince you

there is a relationship between personality traits and zodiac sign? Look back at your answer to

questions 13 and 14 to help you answer this question.

Decision Time!

Now it is time to take a look at the actual class data for Task 1. You will decide if the results provide

evidence that there is a relationship between personality traits and zodiac sign. Fill in your answers:

Choice that best matched your personality: Set that is the correct match for your zodiac sign: Number of students in the class:

Number of students who picked the correct match for their zodiac sign:

Proportion of the class who picked the correct match:

17 Now all the work comes down to this question!

Does the class match proportion provide convincing evidence that personality characteristics

could be related to zodiac sign? Why or why not? Explain your answer using the dotplot in your

reasoning.

NEXT STEPS The tasks you have just completed with the astrology example illustrate a general process that is common to

many statistical investigations. One way to describe this process is as a sequence of four steps, as shown in

the table below.




Steps in a Statistical Investigation





Remember that a statistical investigation is an ongoing process. Often researchers analyze results of one

study and this leads them to think of other research questions. Then, they conduct more research. Also,

researchers start to think more carefully about how they collect data and this may also lead them to think of

ways to improve the data collection process.

18 Identify each step for the astrology investigation to complete the following table.

Steps in a Statistical Investigation For the Astrology Investigation








19 Look at the following two study descriptions. For each study, identify the four steps of the

statistical investigation process to complete the table on the following page.

Study 1

Researchers at the Center for Reproductive Medicine at Brigham and Women’s Hospital wondered

what proportion of women who visit a fertility clinic would want the opportunity to choose the sex

of a future child. They also wondered if those that would like to choose the sex were more likely to

want a boy or girl. The researchers mailed a survey containing 19 questions to women who had

visited the Center. One question on the survey asked women whether they would like the option of

choosing the sex of a future child. If the response to that question was yes, a follow-up question

asked whether they would choose a boy or girl. Of the 229 women who wanted to choose, 89

(38.9%) said they would choose a boy and 140 (61.1%) said they would choose a girl.

Based on their statistical analysis of these data, the researchers concluded that there is convincing

evidence of an overall preference for girls among women wanting to choose the sex of a future

child. The researchers based this conclusion on this fact: In a sample of 229 women, it would be

very unusual to observe a percentage as high as 61.1%. This is unusual if there is really no

preference for girls in the population of women who would like to select the sex of a future child.

Steps in a Statistical Investigation Study 1








Study 2 Psychologists believe that people are less likely to do something if they think it will require a

lot of effort. But, how do people decide what things they think will be hard and what things

they think will be easy?

Researchers at the University of Michigan wondered if written directions would affect how

hard people thought a task would be. If the written instructions for how to do a task were

difficult to read, would that affect how difficult people thought a task was? To investigate

this, they performed an experiment.

The researchers randomly divided twenty students into two groups of 10 students each. One

group received instructions for an exercise routine printed in a font that was easy to read,

and the other group received the same set of instructions printed in a font that was difficult

to read. A sample of each font appears below. Each student read the instructions, and then

they were asked how many minutes they thought the exercise routine would take.

For the group that read the instructions printed in an easy-to-read font, the average number

of minutes they thought the routine would take was 8.23 minutes. For the group that read

the same instructions printed in the hard to read font, the average was 15.1 minutes.

Based on the study data, the researchers concluded that the difference between these two

averages was not likely to be due to chance. There was evidence that people think a task will

be harder when the instructions are hard to read.

This is the easy-to-read font that was used in the study.




This is the hard-to-read font that was used in the study.




Steps in a Statistical Investigation Study 2





20 These two studies both follow the same general process but they are different in some ways.

What are two ways that these studies are different?




TAKE IT HOME In the lesson, your class looked at two different sets of data:

Hypothetical Data: These were the data collected from Task 2, where you were given three

cards (2 cards that had “no match” and 1 that had “match”). These were hypothetical data

because our results were completely by chance. Students picked cards at random, without

reading if it said “match” or “no match.” The “match” proportion was 0.33. Also, the dotplot

the class looked at showed the data on a graph. The hypothetical data allowed you to see

what the proportion and dotplot would look like when there was no relationship between

personality traits and zodiac sign.

Actual Data: These were the data collected from Task 1, where you had to read three sets of

personality traits that were under your birthday.

You compared the hypothetical results to the actual result for the class. You did this to see whether the

proportion in the class was consistent with that chance variation. If the class result was unlikely to occur by

chance alone, this gave some evidence to the astrological theory.

1 Imagine the investigation had given each student a choice of four sets of personality

characteristics for each sign. If there is no relationship between personality characteristics and

sign, about what fraction of the students in the class do you expect to pick the description that

astrologists say matched their sign? Why do you think this? (Remember, the “matching set” is the

set of traits that astrologist say matches a person’s zodiac sign.)

2 There are two dotplots below. One of the dotplots was made by asking 40 students to pick one of

four cards at random. Each card had a set of traits written on it. Researchers computed the

proportion of students that chose the card with the set of traits that matched their zodiac signs by

chance. This process was repeated a large number of times to generate the data used to

construct this one dotplot.

Out of the two dotplots below, which dotplot do you think is the one that was constructed this

way?




3 Imagine that each of these 40 students then made their choices from the list of four personality

types. What proportion of the 40 students need to match correctly to provide convincing evidence

that there is a connection between sign and personality type? Explain your reasoning.

(Helpful hint: Use your answer from Question 2.)

4 Read the following study description:

Researchers at Minnesota State University wanted to learn whether middle-aged adults (ages 40

to 60) who used the Wii Fit video games exercised with enough intensity to meet fitness

recommendations from the United States Government’s report on physical activity. The Wii Fit is

a video game with exercise activities.




The researchers trained 20 middle-aged adults volunteers how to use the Wii Fit video games. On

the day after they were trained, the adults exercised for 20 minutes with the Wii Fit. Researchers

measured the total amount of energy the adults used in calories to be 116 calories for the 20

minute session.

The United States Government recommends that to stay physically fit, middle-aged adults need to

use 150 to 400 calories per day. Based on the results of the study, the researchers concluded the

Wii Fit video games could be a helpful form of exercise for middle aged adults. But, for exercise

with Wii Fit to meet the government’s recommendation, the length of the exercise session should

be increased from 20 minutes to 30 minutes. A Complete the four steps of the statistical investigation process for the study in the table.

Steps in a Statistical Investigation Study





B Based on the results of this study, what next steps do you recommend to the researchers

investigating this issue?




+++++ This lesson is part of STATWAY™, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway™ was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel.

+++++ STATWAY™ and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.

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