AP Stat: Warm-Up Review Chapter 23 #2

Textbook authors must be careful that the reading level of their book is appropriate for the target

audience. Some methods of assessing reading level require estimating the average word length.

Mrs. Wallulis randomly chose 20 words from a randomly selected page in Stats: Modeling the

World and counted the number of letters in each word:

5, 5, 2, 11, 1, 5, 3, 8, 5, 4, 7, 2, 9, 4, 8, 10, 4, 5, 6, 6

1. Suppose that the editor was hoping that the book would have a mean word length of 6.5

letters. Does this sample indicate that the authors failed to meet this goal? Test an appropriate

hypothesis, check condition, show mechanics, and state your conclusion.

2. For a more definitive evaluation of reading level, the editor wants to estimate the text’s mean

word length to within 0.5 letters with 98% confidence. How many randomly selected words

does she need to use?

AP Stat: Warm-Up Review Chapter 24 #1

A total of 23 Gossett High School students were admitted to State University. Of those students,

7 were offered athletic scholarships. The school’s guidance counselor looked at their composite

ACT scores (shown in the table), wondering if State University might admit people with lower

scores if they also were athletes. Assuming that this group of

students is representative of students throughout the state, what do

you think?

1. Test an appropriate hypothesis, check conditions, show

mechanics, and state your conclusion.

2. Create and interpret a 90% confidence interval.

Composite ACT Scores

Non-athletes Athletes

25 21

22 27

19 29

25 26

24 30

25 27

24 26

23 23

22

21

24

27

19

23

17

AP Stat: Chapter 24 Notes

Comparing Means

Comparing two means is not very different from comparing two proportions.

The statistic of interest is the difference in the two observed means, 𝑦1̅̅ ̅ − 𝑦2̅̅ ̅. We need to know

its center, standard deviation and sampling model.

The Center

We laid the groundwork for this a long time ago! If we subtract two random variables, how do

we find the mean of the new resulting random variable? Just subtract the original means!

The Spread

…and what do we do for spread? That’s right—we switch to variance, and we add the individual

variances.

…but since we don’t know σ (since it is a parameter), we must make a substitution. What did we

do in the previous chapter? Let’s do that again. Thus: change σ into s, and take the square root.

The Shape

In the previous chapter, replacing the population standard deviation with the sample standard

deviation forced us to switch distributions—from normal to Student’s t. I wonder what happens

when you switch two population standard deviations…wouldn’t it be cool if the resulting

statistic had a Student’s t distribution, just like before? Alas—for reasons that probably don’t

really interest you (they are complicated)—it doesn’t. The sampling model isn’t really Student’s

t, but only something close. The trick is that by using a special, adjusted degrees-of-freedom

value (as shown below), we can make it so close to a Student’s t model that nobody can tell the

difference.

There are some conditions/assumptions we need to check:

Independence Assumption: The data in each group must be drawn independently and at random

from a homogeneous population so check

Randomization condition: Were the data collected with suitable randomization?

10% condition: We use this condition to make sure we have not violated the

independence assumption by sampling too large a fraction of the population

Normal Population Assumption:

Nearly Normal condition: We must check this for both samples; a violation by either

one violates the condition

A Sampling Distribution for the Difference Between Two Means

When the conditions are met, the sampling distribution of the standardized sample

difference between the means of two independent groups,

𝑡 =(𝑦1̅̅ ̅ − 𝑦2̅̅ ̅) − (𝜇1 − 𝜇2)

𝑆𝐸(𝑦1̅̅ ̅ − 𝑦2̅̅ ̅)

Can be modeled by a Student’s t-model with a number of degrees of freedom found

with a special formula. We estimate the standard error with

𝑆𝐸(𝑦1̅̅ ̅ − 𝑦2̅̅ ̅) = √𝑠12

𝑛1+𝑠22

𝑛2

Independent Groups Assumption: the two groups we are comparing must be independent of

each other

Example 1

A researcher wanted to see whether there is a significant difference in resting pulse rates for men

and women. The data she collected are summarized below.

Gender

Male Female

Count 28 24

Mean 72.75 72.625

Median 73 73

StdDev 5.37225 7.69987

Range 20 29

IQR 9 12.5

Two-sample t-interval

When the conditions are met, we are ready to find the confidence interval for the difference

between means of two independent groups, µ1 - µ2. The interval is (𝑦1̅̅ ̅ − 𝑦2̅̅ ̅) ± 𝑡𝑑𝑓∗ ×

𝑆𝐸(𝑦1̅̅ ̅ − 𝑦2̅̅ ̅) where the standard error of the difference is

𝑆𝐸(𝑦1̅̅ ̅ − 𝑦2̅̅ ̅) = √𝑠12

𝑛1+𝑠22

𝑛2

The critical value 𝑡𝑑𝑓∗ depends on the particular confidence level, C, that you specify and on the

number of degrees of freedom, which we get from the sample sizes and a formula

a) What do the boxplots suggest about any gender differences in pulse rates?

b) Is it appropriate to analyze these data using the methods of inference discussed today?

Explain.

c) Create and interpret 90% confidence interval for the difference in mean pulse rates.

d) Does the confidence interval confirm your answer to (a)? Explain.

Two Sample t-test for the Difference Between Means

The conditions for the two-sample t-test for the difference between the means of two

independent groups are the same as for the two sample t-interval. We test the

hypothesis

𝐻0: 𝜇1 − 𝜇2 = 𝛥0

Where the hypothesized difference is almost always 0, using the statistic

𝑡 =(𝑦1̅̅ ̅ − 𝑦2̅̅ ̅) − ∆0𝑆𝐸(𝑦1̅̅ ̅ − 𝑦2̅̅ ̅)

The standard error of 𝑦1̅̅ ̅ − 𝑦2̅̅ ̅ is

𝑆𝐸(𝑦1̅̅ ̅ − 𝑦2̅̅ ̅) = √𝑠12

𝑛1+𝑠22

𝑛2

When the conditions are met and the null hypothesis is true, this statistic can be

closely modeled by a Student’s t-model with a number of degrees of freedom given

by a special formula. We use that model to obtain a P-Value.

Example 2: A Two-Sample t-test for the Difference Between Two Means

An educator believes that new reading activities for elementary school children will improve

reading comprehension scores. She randomly assigns third graders to an eight-week program in

which some will use these activities and others will experience traditional teaching methods. At

the end of the experiment, both groups take a reading comprehension exam. Their scores are

shown in the back-to-back stem-and-leaf display. Do these results suggest that the new activities

are better? Test an appropriate hypothesis and state your conclusion.

Example 3

A factory hiring people to work on an assembly line gives job applicants a test of manual agility.

This test counts how many strangely shaped pegs the applicant can fit into matching holes in a

one-minute period. The table summarizes the data by gender of the job applicant. Assume that

all conditions necessary for inference are met.

Male Female

Number of subjects 50 50

Pegs placed

Mean:

19.39 17.91

Pegs placed

Std Dev

2.52 3.39

a) Find 95% confidence intervals for the average number of pegs that males and females can

each place.

b) Those intervals overlap. What does this suggest about any gender-based difference in manual

agility?

c) Find a 95% confidence interval for the difference in the mean number of pegs that could be

placed by men and women.

d) What does this interval suggest about any gender-based difference in manual agility?

e) The two results seem contradictory. Which method is correct: doing two-sample inference or

doing one-sample inference twice?

Example 4

The study of a new Core Plus Mathematics Projects methodology tests students’ abilities to solve

word problems. This table shows how the CPMP and traditional groups performed. What do

you conclude?

Math Program n Mean SD

CPMP 320 57.4 32.1

Traditional 273 53.9 28.5

Example 5

Newspaper headlines recently announced a decline in science scores among high school seniors.

In 2000, 15,109 seniors tested by the National Assessment in Education Program (NAEP) scored

a mean of 147 points. Four years earlier, 7537 seniors averaged 150 points. The standard error

of the difference in the mean scores for the two groups was 1.22.

a) Have the science scores declined significantly?

b) The sample size in 2000 was almost double that in 1996. Does this make the results more

convincing, or less? Explain.

Example 6

Some research has been conducted comparing the leg strengths of males and females.

Here are the data (Force, in Newtons) for a random sample of males:

2632 2256 2298 1105 2644 3129 1977

1796 2235 1917 1926 1569 2167

Here are the data for a random sample of females:

1344 1369 1573 1791 1544 1694 1868

1351 2479 1665 1866 2359 2799 2098

Estimate the difference in leg strength between males and females with 99% confidence.