VI 23-1
Part : Learning About the World : Chapters 23 25
Now that we have the principles of inference under our belts, its an easy slide downhill to the
finish line. The mantra of this Part is just like the other ones, only a little different. Emphasize
that, and students wont feel overwhelmed thinking there are new formulas and methods raining
down on them with each chapter.
If you havent done it yet, point your students to the inside back cover of this book. The table there
organizes all of the inference methods and shows them that they are really all alike only a bit
different. And the list of assumptions and conditions should remind students of the things they
were looking for in their displays and summaries back in Parts I and II.
Chapter 23
Inferences for Means
Whats It About?
Chapter 23 examines confidence intervals and hypothesis tests for a population mean. We
introduce the t-models to deal with the extra variation caused by not knowing the populations
standard deviation, and examine the assumptions and conditions necessary for inference. We use
this new application of the concepts of inference to re-examine the correct interpretation of a
confidence interval, the relationship between confidence intervals and hypothesis tests, and the
difference between statistical significance and importance. We show how to determine the sample
size necessary for a study about means, and how to interpret output from software packages.
TI Tips help students explore the new t-models and do the mechanics for confidence intervals and
hypothesis tests about a mean.
Comments
We almost never do inference pretending to know the population standard deviation. We have
emphasized throughout this course that Statistics is about the real world, and we wont change that
now. In the real world no one ever knows the population standard deviation. (Well, almost never.
There are cases when experience suggests that the standard deviation is known the hospital
example in the book is a case in point. In that rare case, we can use the z instead). Gosset
discovered the t-distributions to resolve this dilemma. What worked for Gosset works for us, too.
And we never mention the outdated pretechnology approach of using a Normal model for large
samples. You might think there was a compelling mathematical justification for switching over to
z when the sample size exceeds 30, but there wasnt. Typesetters, not mathematicians, are the
source of that rule of thumb. About 30 lines of a t-table fit conveniently on a page. After that, the
old advice was, turn to the z-table because its pretty close. In the age of graphing calculators and
statistics software, theres no reason to shy away from t for any number of degrees of freedom, so
we can do better than approximate it with z. We can do inference for means the right way, using t.
Always.
The bottom line is that nothing else in this chapter is really new. We still create and interpret
confidence intervals the same old way. We still use the same reasoning and four-step procedure for
testing a hypothesis. The only new issue is the need to scrap the trusty Normal model in favor of a
t-model, an adjustment students will readily make.
Copyright 2010 Pearson Education, Inc.
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Part VI Learning About the World
Weve been very careful about the term standard error, and here we get some of the benefit. The
standard error SE (y ) = s
is a standard error because it is estimated from the data. And, of
n
course, that fact is central to our use of Students t.
Remember also that t-models are just that models. Students most common confusion at this
stage is between the standard deviation of the data, s, and the standard error we estimate for the
mean, SE (y ). They need to be able to distinguish between the actual distribution of the data on the
one hand, and the model for the sampling distribution of the mean on the other. Talk about the
model often and clearly to help them keep that distinction clear.
With the t-models we can show a simpler, clearer relationship between confidence intervals and
hypothesis tests. Students should see these as two ways of viewing essentially the same
calculations. That reduces by half how much new stuff there is here to learn.
Looking Ahead
In the next two chapters we expand our abilities to do inference for means when there are two
groups. Chapter 24 explains the proper procedures if the two groups are independent, and Chapter
25 looks at inference for paired differences if the groups are matched. Both chapters continue to
rely on t-models. Students t returns again in Chapter 27 where we discuss inference for
regression.
Class Dos
Start right in with an example involving data like our coffee machine suggestion. The chapter
discusses confidence intervals first, then moves on to hypothesis tests. You should look at the two
types of inference in that order if you plan to spend two days introducing them.
Its also possible that you can do both in one day a hypothesis test with a follow-up confidence
interval. Students will be able to set up the outline of the hypothesis test. After all, theyre all the
same.
Hypotheses. This time theyre about . (Never, y of course. We know y ; that value comes to
us from the sample. Its the mean of the population we wonder about.)
Model. Some of the conditions are obvious. We need a simple random sample of less than 10%
of the population (and we wont worry as much about the 10% Condition for quantitative data).
Well add another condition after some more discussion.
Mechanics. It seems pretty straightforward at first. Find y , then try to find z. Trouble. The
Central Limit Theorem tells us the sampling model for y has a standard deviation of
divided by the square root of the sample size. Strike One: we dont know . Students suggest
using s, the samples standard deviation, as an estimate. Great idea. This shows they have
caught on to the kind of statistical thinking we want them to develop in this course. But theres
a problem. Unlike proportions, where the null hypothesis specifies p, and that in turn tells us
the SD, here there is no link to the SD. The hypothesized population mean tells us nothing
about the populations standard deviation. And the sample standard deviation, s, varies from
sample to sample. This introduces variation in the denominator, something that wasnt there
before. How can we deal with this extra uncertainty? Some students may suggest cutting
ourselves a little more slack to make up for the increased variability, and thats exactly what a
t-model does.
Copyright 2010 Pearson Education, Inc.
Chapter 23 Inferences About Means
23-3
Spend some time briefly describing t (suggestions follow). Note that the mathematics
underlying the use of t requires the assumption that the sample comes from a population that is
Normal. Strike Two: we dont know the populations center (our hypothesis is about that), we
dont know its spread (thats why we need t), and we dont (indeed cant) know its shape,
either. All we can do is take a look at the distribution of the data in the sample to see if its
plausible that these data were chosen at random from a Normal population. We hope to see a
roughly mound-shaped, symmetric distribution without skewness or outliers. Thats the Nearly
Normal Condition, telling us its OK to use a one-sample t-test for means.
With the conditions now complete, get right on with the hypothesis test. Calculate t, draw the
curve (indicating the degrees of freedom), shade the region representing the P-value, and find
P (using technology or a t table for critical values).
Conclusion. Its the same old story: linkage, decision, and conclusion in context. Be sure the
conclusion talks about the mean of the population.
Its a great idea to follow the hypothesis test with a confidence interval, looking up the critical
value in the table. Spend some time with the interpretation. There are all kinds of new ways to be
wrong about what the interval means! Most common are that some students will think that 95% of
the population or 95% of all sample means must lie in the interval. Help students understand the
correct interpretation. The Investigative Task on means will help.
Since the t-models are new, it helps to discuss their relationship to the Normal model. Common
sense says that the increased uncertainty caused by using s to estimate makes this model more
spread out, and that this uncertainty (and therefore the spread) must be greater for small samples.
These ideas tell us that there are different t-models for different ns, pointing to the new issue of
degrees of freedom. Greater spread tells us that confidence intervals must be wider, so critical
values of t must be larger than the corresponding critical value of z. A quick glance down one of
the columns of the table will show that, and also show that t-models look increasingly Normal as n
increases.
The Nearly Normal Condition is intentionally vague. First, be sure students understand that the
issue is not whether the sample is Normal. Samples arent; they cant be. We must have a sample
drawn from a population thats Normal, but we can never be sure about the population. The t-test
is fairly robust. What we really need is that the distribution is roughly symmetric i.e., nearly
Normal. Were particularly concerned about outliers and skewness. However, the usefulness of a tmodel also depends on the sample size. As the sample size increases, the Normality of the
population becomes less critical. (We say that the t-models with larger degrees of freedom are
increasingly robust with respect to the assumption of Normality.) For ns near 20, some skewness
is okay. For ns near 30, we can use t unless the sample is strongly skewed or there are far outliers.
For larger samples, we often dont even look at the histogram, justifying use of a t-model simply
by saying the sample is large.
Talk about what to do with outliers. We recommend performing the analysis twice. The concern,
of course, is that the outlier will distort the results. If the conclusions dont change (if, for example,
we reject a hypothesis when we include the outlier in the data, and also reject it after we remove
the outlier), then its clear that the decision is not based on the impact of the outlier. Generally,
outliers inflate the standard error more than they move the sample mean, so they make confidence
intervals wider and P-values larger.
Copyright 2010 Pearson Education, Inc.
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Part VI Learning About the World
We resort to z only when we do a sample size calculation, because not knowing n leaves us unable
to pick a critical value for t. This calculation will always underestimate the sample size we should
use. If it calls for a very large sample, thats fine. But if it points to a small sample, then we need to
use that first estimate for n to determine a critical t and rerun the calculation.
The Importance of What You Dont Say
Dont suppose we know , the standard deviation of the population. We (almost) never will.
Dont suggest that we use z instead of t for large samples. For inference about means, use t.
Always. (Almost.)
Dont confuse notation or terminology. We find it confusing to write y = / n . Instead use
SD( y ). Of course its the same thing, but students seem to get confused with s all over the
place. And the difference between SD( y ) and SE( y ) reminds them of the difference in
terminology and meaning.
A Calculator Note
Newer TI-84 operating systems allow students to find critical t values with InvT. Students with TI83s will need to use the table. (Table T) Or they can trick the TI into doing the same thing.
Suppose you want a 95% confidence interval for 10 df. Use 8:Tinterval. Enter x : 0, Sx: (11) ,
n:11, and C-level:.95. The calculated confidence interval will be the critical values of t. (You
may or may not want to show this to your students. After all, if they were going to use the TI
instead of the table to get the critical value, why not just have it do the interval?)
Class Examples
1. A coffee machine dispenses coffee into paper cups. Youre supposed to get 10 ounces of
coffee, but the amount varies slightly from cup to cup. Here are the amounts measured in a
random sample of 20 cups. Is there evidence that the machine is shortchanging customers?
9.9
9.9
10.0
9.9
10.0
Hypotheses. H0 : = 10.0
9.7
9.6
9.5
9.6
9.9
10.0
9.8
9.7
10.2
9.5
10.1
9.8
10.1
9.8
9.9
HA : < 10.0
Model. Random sample; 20 < 10% of all cups (no reason to doubt
independence). The histogram of sample data to the right looks roughly
unimodal and symmetric, so its reasonable to believe that the amount of
coffee in all possible cups could be described by a Normal model. Okay
to do a t-test for the mean.
Mechanics. n = 20 df = 19 y = 9.845 s = 0.199
9.845 10
t=
= 3.49
0.199
20
P = P(t19 < 3.49) = 0.0012
Copyright 2010 Pearson Education, Inc.
Chapter 23 Inferences About Means
23-5
Conclusion. Such a small P-value makes it unlikely that the low sample mean resulted from
sampling error, so we reject the null hypothesis. There is strong evidence that the mean amount
of coffee dispensed by this machine is less than the stated 10 fluid ounces.
Confidence interval. The conditions have been met, so we can create a one-sample t-interval.
(Note that all confidence intervals look alike: estimate margin of error.)
*
y t19 SE ( y ) = 9.845 2.093
0.199
= (9.75,9.94)
20
We are 95% confident that the machine dispenses an average of between 9.75 and 9.94 fluid
ounces of coffee per cup.
2. A company has set a goal of developing a battery that lasts over 5 hours (300 minutes) in
continuous use. A first test of 12 of these batteries measured the following lifespans (in
minutes): 321, 295, 332, 351, 281, 336, 311, 253, 270, 326, 311, and 288.
Is there evidence that the company has met its goal?
Solution:
We want to know if the mean battery lifespan exceeds the 300-minute goal set by the
manufacturer. We have 12 battery lifespans in our sample to test the claim.
Hypotheses. The null hypothesis is that the batteries have a mean lifespan of 300 minutes.
The 300-minute goal has not been met. The alternative hypothesis is that the batteries have
a mean lifespan greater than 300 minutes. The 300-minute goal has been met.
H 0 : = 300, H A : > 300
Model.
4
Randomization Condition: This is not a random sample of
batteries, but merely 12 batteries produced for preliminary
3
testing. However, it is reasonable to assume that these
2
batteries are representative of all batteries.
1
Nearly Normal Condition: The distribution of battery
lifespans is roughly unimodal and symmetric, so its
240
280
320
reasonable to assume that the lifespans of all batteries could
Battery Lifespans (min)
be described by a Normal model.
Since the conditions have been met, we can do a one sample t-test for the mean, with 11
degrees of freedom.
Mechanics.
t=
n = 12
y
s
n
306.25 300
t=
29.31
df = 11
y = 306.25
P-value = P( y > 306.25)
= P(t11 > 0.7387)
= 0.238
12
t 0.7387
Copyright 2010 Pearson Education, Inc.
s = 29.31
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Part VI Learning About the World
Conclusion. Since the P-value is high, we fail to reject the null hypothesis. There is no
evidence to suggest that the mean battery lifespan exceeds 300 minutes. It does not appear
that the company has met its goal.
Find a 90% confidence interval for the mean lifespan of this type of battery.
Solution. The conditions have been met, so we can create a one-sample t-interval, with
90% confidence.
29.31
*
y t11 SE ( y ) = 306.25 1.796
= (291.05,321.45)
12
I am 90% confident that the mean battery lifespan is between 291.05 and 321.45 minutes.
If we wish to conduct another trial, how many batteries must we test to be 95% sure of
estimating the mean lifespan to within 15 minutes? To within 5 minutes?
Solution: We want to know how many batteries to test to be 95%
sure of estimating the mean lifespan to within 15 minutes. First, do
a preliminary estimate using z * = 1.96 as the critical value. Our
first estimate is about 15 batteries, a small sample.
s
ME = z *
n
29.31
15 = 1.96
n
(1.96)(29.31)
15
n 14.67
n=
*
Now, do a better estimate, using t14 = 2.145 as the critical value.
s
ME = t *
n
29.31
15 = 2.145
n
We would need to sample about 18 batteries in order to
estimate the mean battery lifespan to within 15 minutes, with
95% confidence.
(2.145)(29.31)
15
n 17.56
n=
Finally, to estimate the mean battery lifespan to within 5 minutes, you could do the entire
process again, perhaps using a critical value with much higher degrees of freedom. We
know that its going to take lots more batteries to cut the margin of error to a third of what
it was. Alternatively, we know it will take a sample about 9 times as large, 18(9) = 162
batteries, since the margin of error was decreased to a third of its size. (The standard error
involves the square root of the sample size in its denominator.)
Copyright 2010 Pearson Education, Inc.
Chapter 23 Inferences About Means
23-7
3. A few days after you finish this chapter you will be collecting (and then returning) the Tasks.
Each student will have created a confidence interval based on a different sample. Tell them
what the true mean for the population was. (If you used our data, the mean Math SAT-I score
was 611.32.) Ask them how many caught it in their interval. Then ask who missed it. There
will probably be a few whose intervals do not contain the true mean yet they did nothing
wrong. This teachable moment really helps students understand what 95% confidence
means, and why we are so picky about those pesky interpretations.
Resources
ActivStats
Chapter 23. The interactive tool in ActivStats that shows confidence intervals and P-values
dynamically is valuable in cementing student understanding. One activity in this chapter
differentiates the effects of smaller sample size on degrees of freedom from the effects of
sample size on the standard error (in which we divide by the sample size). The change in
standard error has a far greater impact than the spreading of the t-distribution with smaller
degrees of freedom.
TI-Nspire Demonstrations
The t-models
o Understand the t-models.
o See how t-models change as you change the degrees
of freedom.
o Compare t-models to the Normal model.
Confidence Intervals for Means
o Generate confidence intervals from many samples how to
see often they successfully capture the true
mean.
Decisions Through Data
Video Unit 20: Confidence Intervals done the old-fashioned way, unfortunately,
assuming we know the populations standard deviation. There are some nice parts of this
video clip, but dont show it if you fear the approach will confuse your students.
Video Unit 21: Tests of Significance
Workshop Statistics
Topic 20: Confidence Intervals II: Means
Topic 22: Tests of Significance II: Means
Topic 23: More Inference Considerations
Copyright 2010 Pearson Education, Inc.
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Part VI Learning About the World
Web Links
Youll find several good data sets at the DASL Web site. (lib.stat.cmu.edu/DASL)
Several Web sites have confidence interval applets, useful for demonstrating the key ideas
that the population parameter is fixed while confidence intervals vary from sample to
sample, capturing the population value most but not all of the time.
Assignments
Two days of reading and three assignments of 56 exercises each will suffice for this chapter.
Over three days you can do a couple of examples in class and go over homework questions (with
time to look at the last unit test). Students will then be ready to tackle the Task as you move on
into Chapter 24. Three chapter quizzes are provided.
Investigative Task
This is the first of a pair of Tasks that use the same data set. The thrust of the Tasks is more
important than the particular data you use. We use the Math and Verbal SAT scores reported for
one high school, but other data will work. You may want to find something more relevant to your
students and revise the Task accordingly. Ideally you need a reasonably large but manageable data
set; 250 to 500 cases would work well. Math and Verbal SATs (or ACTs) are ideal because each
individual has two scores (making paired comparisons possible) and lots of statistics are available.
The first part of the Task asks students to select a random sample, reviewing use of random
numbers and sampling issues from Chapter 12. Based on the various samples, students construct
confidence intervals, and then use their interval to compare the local performance to state or
national results.
And this affords you the opportunity to look at all their confidence intervals together, noting that
most hit the target while others miss.
The second in this pair of these Tasks comes after Chapter 25, asking students to do two
hypothesis tests one involving paired data and the other comparing two independent means.
Have them hang on to the data sheet; theyll need it again.
Copyright 2010 Pearson Education, Inc.
AP Statistics Quiz A Chapter 23
Name
A professor at a large university believes that students take an average of 15 credit hours per term.
A random sample of 24 students in her class of 250 students reported the following number of
credit hours that they were taking:
12 13 14 14 15 15 15 16 16 16 16 16
17 17 17 18 18 18 18 19 19 19 20 21
1. Does this sample indicate that students are taking more credit hours than the professor
believes? Test an appropriate hypothesis and state your conclusion.
2. Find a 95% confidence interval for the number of credit hours taken by the students in the
professors class. Interpret your interval.
23-9
Copyright 2010 Pearson Education, Inc.
AP Statistics Quiz A Chapter 23 Key
1. H 0 : = 15 . The mean number of credit hours taken by all of
5
the professors students is 15 credit hours.
4
H A : > 15 The mean number of credit hours taken by all of
3
the professors students is greater than 15 credit hours.
2
* Randomization condition: Students from the class were
1
randomly sampled.
* 10% condition: The sample is less than 10% of the class
12
15
18
21
population.
Credit Hours
* Nearly Normal condition: The histogram of credit hours is
unimodal and reasonably symmetric.
Under these conditions, the sampling distribution of the mean can be modeled by Students t
with df = n 1 = 24 1 = 23. Use a one-sample t-test for the mean.
We know: n = 24, y = 16.6 , and s = 2.22 .
2.22
So, SE ( y ) =
= 0.453 .
24
y 0 16.6 15
=
= 3.532 .
t=
0.453
SE ( y )
The P-value is P(t23 > 3.532) = 0.0009 .
The P-value of 0.0009 is low, so reject the null hypothesis. There is strong evidence that the
professors students are taking more than 15 credit hours, on average.
2. With the conditions satisfied (from Problem 1), we can find a t-interval for mean credit hours.
We know: n = 24, y = 16.6 , s = 2.22 , and SE ( y ) =
Our confidence interval has the form y tn 1
interval is then 16.6 2.069
2.22
= 0.453 .
24
s
. We have t23 = 2.069 . Our 95% confidence
n
2.22
= 16.6 0.94 , or 15.66 to 17.54.
24
We are 95% confident that the interval 15.66 to 17.54 contains the true mean number of credit
hours that students in the professors class are taking.
23-10
Copyright 2010 Pearson Education, Inc.
AP Statistics Quiz B Chapter 23
Name
Insurance companies track life expectancy information to assist in determining the cost of life
insurance policies. The insurance company knows that, last year, the life expectancy of its
policyholders was 77 years. They want to know if their clients this year have a longer life
expectancy, on average, so the company randomly samples some of the recently paid policies to
see if the mean life expectancy of policyholders has increased. The insurance company will only
change their premium structure if there is evidence that people who buy their policies are living
longer than before.
86 75 83 84 81 77 78 79 79 81
76 85 70 76 79 81 73 74 72 83
1. Does this sample indicate that the insurance company should change its premiums because life
expectancy has increased? Test an appropriate hypothesis and state your conclusion.
2. For more accurate cost determination, the insurance companies want to estimate the life
expectancy to within one year with 95% confidence. How many randomly selected records
would they need to have?
23-11
Copyright 2010 Pearson Education, Inc.
AP Statistics Quiz B Chapter 23 Key
1. H 0 : = 77 years . The mean life expectancy of the companys patients is 77 years.
H A : > 77 years . The mean life expectancy of the
3
companys patients is greater than 77 years.
2
* Randomization condition: The records from the insurance
company were randomly sampled.
1
* 10% Condition: 20 records represent less than 10% of the
companys records.
70
74
78
82
86
* Nearly Normal condition: The histogram of the ages at death
Age at Death
is unimodal and reasonably symmetric. This is close enough
to Normal for our purposes.
Under these conditions, the sampling distribution of the mean can be modeled by Students t
with df = n 1 = 20 1 = 19. We will use a one-sample t-test for the mean.
We know: n = 20, y = 78.6 years, and s = 4.48 years.
s
4.48
SE ( y ) =
=
= 1.002 years.
n
20
y 0 78.6 77
=
= 1.597 .
t=
SE ( y )
1.002
P = P(t19 > 1.597) = 0.063
The P value of 0.063 is fairly high, so we fail to reject the null hypothesis. The insurance
company shouldnt need to increase their premiums because there is little evidence to indicate
that people who buy their policies are living longer than before.
2. We wish to find the sample size, n, that would allow a 95% confidence level for the mean life
expectancy of a policy holder from the insurance company to have a margin of error of only
one year.
First estimate:
ME = z * SE ( y )
4.48
n
n = 77.1 78
1 = 1.96
Although not necessary, since 78 is quite large, we could find a
*
better estimate using t75 = 1.992 , from Table T.
*
ME = t75 SE ( y )
4.48
n
n = 79.6 80
1 = 1.992
23-12
Copyright 2010 Pearson Education, Inc.
AP Statistics Quiz C Chapter 23
Name
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.
Weve randomly chosen 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 our 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 and state your conclusion.
2. For a more definitive evaluation of reading level the editor wants to estimate the texts mean
word length to within 0.5 letters with 98% confidence. How many randomly selected words
does she need to use?
23-13
Copyright 2010 Pearson Education, Inc.
AP Statistics Quiz C Chapter 23 Key
1. H 0 : = 6.5 . The mean word length in the book is 6.5 words.
H A : 6.5 . The mean word length in the book is not 6.5 words.
* Randomization Condition: We have a random sample of the
8
words in the book.
* 10% Condition: The words are less than 10% of all words in 6
the book.
4
* Nearly Normal Condition: A histogram of the observed
2
word lengths looks roughly unimodal and symmetric, so the
population of all word lengths may be approximately
Normal.
Since the conditions are satisfied, we will to use a two-tail,
one-sample t-test.
y = 5.5
n = 20
s = 2.685
df = 19
5.5 6.5
t=
= 1.67
2.865
20
P = 2 P(t19 < 1.67) = 0.11
0
6
Word Length (letters)
Because the P-value is so high we do not reject the null hypothesis. This sample does not
provide evidence that the average word length differs from the goal of 6.5 letters.
2. First estimate:
ME = z * SE ( y )
2.685
n
n = 156.02 157
0.5 = 2.326
Although not necessary, since 157 is quite large, we could find a
*
better estimate using t140 = 2.353 , from Table T.
*
ME = t140 SE ( y )
2.685
n
n = 159.66 160
0.5 = 2.353
23-14
Copyright 2010 Pearson Education, Inc.
12
AP Statistics Investigative Task
Chapter 23
SAT Performance
This is the first of two related Tasks. In these Tasks you will investigate four questions about
students SAT scores at one high school.
1. What is the mean SAT-Math score at this high school?
2. How do these students SAT Math scores stack up against the rest of the state?
3. Is there a significant difference between Verbal and Math scores for these students?
4. Nationally males tend to have higher Math scores than females. Is that true at this high
school, too?
You have a copy of data from the SAT Score Roster sent to the high school by the College Board.
It shows gender, Verbal scores, and Math scores, for 303 students in a recent graduating class.
NOTE: You will not use all of these data, just a sample of 20 30 students.
Your current assignment: investigate the first two questions posed above.
Draw a sample of these students, carefully explaining your procedure.
Use your sample to create a 95% confidence interval for the mean SAT-I Math score at this
high school.
Based on your confidence interval, compare the performance of these students with the
statewide mean Math score of 503.
Prepare a written report that includes complete demonstrations of the statistical procedures
you use and your conclusions (in context, of course).
Important: Save the Score Roster and your sample. Youll be using them to answer the other two
questions in the next Investigative Task coming soon to a Statistics class near you.
23-15
Copyright 2010 Pearson Education, Inc.
AP Statistics Investigative Task
Chapter 23
This task has 4 components; each is scored as Essentially correct, Partially correct, or Incorrect.
1. The Sample
E Selects a random sample, explains the sampling process clearly, calculates the correct
summary statistics (perhaps with minor arithmetic errors), and uses the proper vocabulary
and notation.
P Selects a random sample, but explanation of the process may be unclear or there may be
mistakes in notation, or vocabulary.
I Sample is not random or the process is not explained or there are several major mistakes in
arithmetic, notation, or vocabulary.
2. The Conditions
E Cites randomness, <10% of all possible students, and checks normality with a plot.
P Discusses normality but omits the plot or fails to mention one of the other conditions.
I Misunderstands or omits the conditions, or lists irrelevant issues (np 10).
3. The Mechanics
E Identifies the procedure, shows the sample statistics and degrees of freedom, writes the
formula using correct critical value and notation, and calculates the correct interval
(perhaps with minor arithmetic or rounding errors).
P Appears to be doing the proper procedure but omits important information, uses the wrong
critical value, uses the wrong notation, or makes major errors in calculations.
I Uses the wrong procedure or shows no work or makes several major mistakes.
4. The Interpretation
E Correctly interprets the confidence interval in the proper context, and compares the local
performance to statewide results.
P Writes a conclusion that is correct but not in context or doesnt compare local performance
to statewide results.
I Does not interpret the confidence interval correctly.
Comments:
Scoring
Es count 1 point, Ps are 1/2
AP score = sum of 4 components; rounding based on quality of P responses
Grade: A = 4, B = 3, etc., with +/- based on rounding (ex: 3.5 rounded to 3 is a B+)
Name _______________________________
AP Score ____ Grade ____
23-16
Copyright 2010 Pearson Education, Inc.
NOTE: We present a model solution with some trepidation. This is not a scoring key, just an
example. Many other approaches could fully satisfy the requirements outlined in the scoring
rubric. That (not this) is the standard by which student responses should be evaluated.
Model Solution - Investigative Task SAT Performance
I want to determine the mean SAT-Math score at this high school. I will take a simple random
sample of 25 students by assigning a number 001-300 to each of the students, and then choosing
25 random numbers 001-300, ignoring repeats.
I want to find a 95% confidence interval for the mean SAT-Math score of all students at this high
school. I have data on the scores of 25 students, from a simple random sample of the 300 students
at this high school.
Model.
10
Randomization Condition: The students were chosen by a simple
random sample.
10% Condition: 25 students represent less than 10% of the
population of 300 students.
8
6
4
2
Nearly Normal Condition: The distribution of SAT-Math scores
in the sample is unimodal and symmetric.
375
600
Math
The conditions are satisfied, so I will use a Students t-model with (n 1) = 25 1 = 24 degrees of
freedom to find a one-sample t-interval for the mean.
Mechanics. From my sample of 25 students: n = 25 scores
y = 602.8
s = 74.3034
74.3034
*
y t24 SE ( y ) = 602.8 2.064
= 602.8 30.672 = (572.13, 633.47)
25
Conclusion. I am 95% confident that the interval from 572.13 to 633.47 contains the true mean
SAT-Math score for students at this high school.
According to my confidence interval, students at this high school had a higher mean SAT-Math
score than students nationwide. The national mean of 503 was not included in my 95% confidence
interval.
23-17
Copyright 2010 Pearson Education, Inc.
23-18
Copyright 2010 Pearson Education, Inc.
AP Statistics SAT Data
Gender
F
F
M
M
M
M
F
M
F
F
M
F
F
M
M
M
F
M
F
M
M
M
F
F
F
M
M
F
F
M
F
F
M
F
M
F
F
F
M
M
M
M
M
M
M
F
F
F
F
F
F
F
Verbal
450
640
590
400
600
610
630
660
660
590
580
500
480
530
690
480
450
650
620
630
490
560
710
360
590
520
510
630
740
680
660
690
540
670
580
600
600
710
640
490
650
580
710
740
510
650
580
600
670
640
370
620
Math
450
540
570
400
590
610
610
570
720
640
650
540
360
520
640
570
420
650
600
740
520
620
720
390
530
630
600
630
760
690
700
750
560
520
720
560
560
640
650
590
680
710
730
700
560
760
660
640
700
670
460
470
Gender
M
F
M
M
F
M
F
M
M
M
M
M
M
M
F
F
M
M
M
F
M
F
M
F
M
M
M
F
M
M
M
M
F
F
M
F
F
F
F
F
M
F
M
M
M
F
F
F
M
F
F
F
Verbal
680
620
590
430
680
650
650
660
640
460
430
460
800
310
510
770
750
580
410
430
630
630
480
420
800
650
580
560
670
610
510
560
770
530
650
730
690
610
710
730
450
400
700
580
600
630
600
630
690
630
480
680
Math
650
620
690
540
740
650
680
700
680
680
480
600
720
530
550
700
670
630
390
400
680
630
550
430
650
500
690
540
760
740
610
530
690
430
740
790
640
510
680
590
570
410
620
660
670
690
640
700
740
560
500
630
Gender
F
M
M
M
M
F
F
M
M
F
F
M
M
F
F
F
F
M
F
F
F
F
F
F
F
M
F
M
F
M
M
M
F
M
F
M
F
F
M
M
M
M
F
F
M
F
F
F
M
F
F
M
Verbal
650
460
560
610
620
590
390
510
450
520
470
690
510
770
520
550
680
740
660
740
730
560
570
560
670
650
690
610
500
560
640
430
700
620
610
580
730
520
540
640
680
580
640
700
600
540
480
710
650
640
370
710
Math
680
410
560
760
650
640
440
530
440
690
410
620
540
710
570
600
580
700
690
640
680
630
530
540
520
710
700
740
650
700
650
490
570
670
640
640
570
530
580
610
720
490
630
650
630
510
540
700
780
570
410
700
23-19
Copyright 2010 Pearson Education, Inc.
Gender
F
F
M
M
M
F
F
M
F
M
M
F
M
F
M
F
F
M
F
M
M
M
F
M
F
M
M
M
F
F
M
F
F
M
M
M
M
F
M
M
F
M
F
M
F
M
M
F
F
M
F
F
F
F
Verbal
630
590
750
600
610
490
680
520
680
650
600
550
490
530
560
630
510
710
550
690
700
540
280
710
640
600
610
680
520
730
510
620
530
550
590
620
640
490
530
560
710
480
490
460
630
510
600
550
670
560
600
650
630
440
Math
660
580
800
690
550
800
610
540
660
700
560
560
390
530
560
590
520
740
560
620
700
620
500
760
710
590
670
670
470
740
680
740
500
680
670
640
570
530
580
600
700
530
490
560
520
520
610
570
580
520
600
530
610
470
Gender
M
M
M
M
M
M
F
M
F
F
F
M
M
M
M
F
M
F
M
M
M
F
F
F
M
M
M
F
F
M
F
F
M
M
F
F
F
M
F
F
F
F
F
M
M
F
M
F
M
F
M
M
M
M
Verbal
650
690
360
510
610
700
540
600
610
700
490
390
580
760
730
590
640
610
760
570
700
630
530
490
600
580
710
600
560
570
630
640
550
690
530
580
550
690
420
690
560
560
490
430
690
600
700
630
620
550
600
450
690
590
Math
560
670
290
510
570
740
490
670
710
570
560
630
530
760
760
620
740
620
700
630
740
630
670
700
660
760
700
450
590
690
610
570
600
670
490
640
560
740
410
720
570
560
510
570
670
550
630
710
610
550
670
610
680
600
23-20
Copyright 2010 Pearson Education, Inc.
Gender
M
F
M
M
M
F
M
M
F
F
F
M
M
M
F
M
F
M
M
M
M
M
F
F
F
F
F
F
M
F
F
M
M
F
M
M
M
M
M
Verbal
760
650
540
580
670
650
660
500
610
580
570
690
570
600
540
640
680
730
610
500
580
620
570
690
610
630
540
760
470
340
480
690
620
710
440
400
650
600
700
Math
580
600
630
530
650
680
660
580
510
500
620
620
540
670
670
760
600
670
550
550
630
710
630
610
570
570
660
690
470
450
500
760
610
630
740
710
740
660
570