Post on 18-Mar-2021
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.