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Review and Practice for Quizzes and Tests

UNIT 6 Wrap-Up

PART I
®
FRAPPY! Free Response AP Problem, Yay!
Take advantage of the review tools included at the end of every unit
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and part. Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correct-
ness of your methods as well as on the accuracy and completeness of your results and explanations.

Members at a popular fitness club currently pay a $40 per month membership fee. The owner of the club
wants to raise the fee to $50 but is concerned that some members will leave the gym if the fee increases.
To investigate, the owner plans to survey a random sample of the club members and construct a 95% con-
fidence interval for the proportion of all members who would quit if the fee was raised to $50.
UNIT 6 Wrap-Up (a) Explain the meaning of “95% confidence” in the context of the study. ± 0.075 . Inter-
DO THE FRAPPY! Learn how to answer

(b) After the owner conducted the survey, he calculated the confidence interval to be 0.18

PART I pret this interval in the context of the study. FRQs successfully by working the FRAPPY! —
®
the Free Response AP Problem, Yay! — that
(c) According to the club’s accountant, the fee increase will be worthwhile if fewer than 20% of the mem-

bers quit. According to the interval from part (b), can the owner be confident that the fee increase will
®
be worthwhile? Explain.
FRAPPY! Free Response AP Problem, Yay! begins each Unit/Part Wrap-up.
(d) One of the conditions for calculating the confidence interval in part (b) is that np ˆ ≥10 and

n (1 − ˆ ) p ≥10 . Explain why it is necessary to check this condition. UNIT 6, PART I Review 601

Find the P-value by calculating the probability of getting
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correct- true. Besides helping you draw a conclusion, the interval tells
a z statistic this large or larger in the direction specified
ness of your methods as well as on the accuracy and completeness of your results and explanations. UNIT 6, PART I Review 601 you which alternative parameter values are plausible.
After you fi nish the FRAPPY!, you can view two example solutions on the book’s website (bfwpub.com/tps7e).
by the alternative hypothesis H a in the standard normal Because conclusions are based on sample data, there
Determine whether you think each solution is “complete,” “substantial,” “developing,” or “minimal.” If the
distribution. If you are performing a two-sided test, make
solution is not complete, what improvements would you suggest to the student who wrote it? Finally, your
Members at a popular fitness club currently pay a $40 per month membership fee. The owner of the club is a possibility that the conclusion to a significance test will
sure to find the area in both tails of the standard normal
wants to raise the fee to $50 but is concerned that some members will leave the gym if the fee increases. be incorrect. You can make two types of errors: A Type I
teacher will provide you with a scoring rubric. Score your response and note what, if anything, you would do
Find the P-value by calculating the probability of getting true. Besides helping you draw a conclusion, the interval tells error occurs if you find convincing evidence for the alterna-
distribution.
To investigate, the owner plans to survey a random sample of the club members and construct a 95% con-

differently to improve your own score.
Whenever you are asked if there is convincing evidence
a z statistic this large or larger in the direction specified you which alternative parameter values are plausible. tive hypothesis when, in reality, the null hypothesis is true.
fidence interval for the proportion of all members who would quit if the fee was raised to $50.
A Type II error occurs when you don’t find convincing evi-
for a claim about a population parameter, you are expected
by the alternative hypothesis H a in the standard normal Because conclusions are based on sample data, there dence that the alternative hypothesis is true when, in reality,
to respond using the familiar four-step process.
(a) Explain the meaning of “95% confidence” in the context of the study.

UNIT 6, PART I REVIEW
distribution. If you are performing a two-sided test, make is a possibility that the conclusion to a significance test will the alternative hypothesis is true. The probability of making
State: State the hypotheses, parameter(s), and signifi-
(b) After the owner conducted the survey, he calculated the confidence interval to be 0.18

± 0.075 . Inter-
cance level.
sure to find the area in both tails of the standard normal be incorrect. You can make two types of errors: A Type I a Type I error is equal to the significance level (α) of the test.
pret this interval in the context of the study.

Plan: Identify the appropriate inference method and
Decreasing the probability of a Type I error increases the
distribution. error occurs if you find convincing evidence for the alterna- size from the same population and used them to construct
probability of a Type II error, and increasing the probability

check the conditions.
SECTION 6A Confidence Intervals: The Basics
(c) According to the club’s accountant, the fee increase will be worthwhile if fewer than 20% of the mem-

Whenever you are asked if there is convincing evidence tive hypothesis when, in reality, the null hypothesis is true. C % confidence intervals, about C % of those intervals would
of a Type I error decreases the probability of a Type II error.
Do: If the conditions are met, perform calculations.
bers quit. According to the interval from part (b), can the owner be confident that the fee increase will ned that a point estimate is the
In this section, you lear
for a claim about a population parameter, you are expected A Type II error occurs when you don’t find convincing evi- capture the [parameter in context].”
The probability that you avoid making a Type II error
• Calculate the test statistic.

single best guess for the value of a population parameter.
be worthwhile? Explain.

SNAPSHOT REVIEW Study the Unit
• Find the P-value.
when an alternative value of the parameter is true is called the
to respond using the familiar four-step process. dence that the alternative hypothesis is true when, in reality, SECTION 6B Confidence Intervals for a

You also learned that a confidence interval, also known as
ˆ
power of the test. Power is good — if the alternative hypothesis
Conclude: Make a conclusion about the hypotheses in

(d) One of the conditions for calculating the confidence interval in part (b) is that np

≥10 and
State: State the hypotheses, parameter(s), and signifi-
an interval estimate, provides an interval of plausible values
Review, which gives a short summary of the alternative hypothesis is true. The probability of making is true, you want to maximize the probability of finding con-
Population Proportion

the context of the problem.

n
− ˆ )
(1
≥10 . Explain why it is necessary to check this condition.
p
for a parameter based on sample data. To interpret a confi-
cance level.
a Type I error is equal to the significance level (α) of the test.
You can also use a confidence interval to make a conclu-
vincing evidence that it is true. We can increase the power of
dence interval, say, “We are C % confident that the interval
each section, to be sure you understand Decreasing the probability of a Type I error increases the the In this section, you learned how to construct and interpret
Plan: Identify the appropriate inference method and
a significance test by increasing the sample size, by increasing
sion for a two-sided test. If the null parameter value is one of
captures the
to

confidence intervals for a population proportion. Three con-
from

the significance level, and by reducing the standard error with
plausible values in the interval, there isn’t convincing evidence
check the conditions.
[parameter in context],” where C is the confidence level of the
After you fi nish the FRAPPY!, you can view two example solutions on the book’s website (bfwpub.com/tps7e).
that the alternative hypothesis is true. However, if the interval
wise data collection methods. The power of a test will also be
the key concepts in each section. probability of a Type II error, and increasing the probability ditions must be met to ensure that the observations in the sam-
of a Type I error decreases the probability of a Type II error.
Do: If the conditions are met, perform calculations.
ple are independent and that the sampling distribution of ˆ p
interval. You can use a confidence interval to evaluate a claim
Determine whether you think each solution is “complete,” “substantial,” “developing,” or “minimal.” If the
contains only values consistent with the alternative hypothesis,
greater when the alternative value of the parameter is farther
The probability that you avoid making a Type II error
• Calculate the test statistic.
about the value of a population parameter.
solution is not complete, what improvements would you suggest to the student who wrote it? Finally, your is approximately normal. First, the data used to calculate the
there is convincing evidence that the alternative hypothesis is
away from the null hypothesis value.
interval must come from a random sample from the popula-
The confidence level C describes the percentage of
• Find the P-value.
SUMMARY TABLES in Units 5–9 review when an alternative value of the parameter is true is called the tion of interest (the Random condition). When the sample is
teacher will provide you with a scoring rubric. Score your response and note what, if anything, you would do

confidence intervals that we expect to capture the value

differently to improve your own score.
Conclude: Make a conclusion about the hypotheses in
power of the test. Power is good — if the alternative hypothesis
of the parameter in repeated sampling. To interpret a C %
Inference for a Population Proportion
important details of each sampling is true, you want to maximize the probability of finding con- selected without replacement from the population, the sample
the context of the problem.
confidence level, say, “If we took many samples of the same
size should be less than 10% of the population size (the 10%
602
UNIT 6 Inference for Categorical Data: Proportions
Confidence Interval for p
Significance Test for p
You can also use a confidence interval to make a conclu-
distribution and inference procedure, vincing evidence that it is true. We can increase the power of One-sample z test for p (1-PropZTest)
UNIT 6, PART I REVIEW
Name (TI-83/84) One-sample z interval for p (1-PropZInt)
sion for a two-sided test. If the null parameter value is one of the a significance test by increasing the sample size, by increasing
Null Hypothesis
Not applicable.
plausible values in the interval, there isn’t convincing evidence the significance level, and by reducing the standard error with : 0 H p = 0 p Related Example Relevant Unit
including conditions and formulas.

• Random: The data come from a random sample from
SECTION 6A Confidence Intervals: The Basics
Conditions

size from the same population and used them to construct
that the alternative hypothesis is true. However, if the interval wise data collection methods. The power of a test will also be • Random: The data come from a random sample from Section on Page(s) Review Exercise(s)
Learning Target
the population of interest.
the population of interest.
C % confidence intervals, about C % of those intervals would
contains only values consistent with the alternative hypothesis, greater when the alternative value of the parameter is farther 0.10N . Check the conditions for calculating a confidence interval for a population 6B 541 R2
In this section, you learned that a point estimate is the

10%: When sampling without replacement, n <
0.10N .
10%: When sampling without replacement, n <

capture the [parameter in context].”
07_StarnesTPS7e_40934_un06_529_642.indd 599
09/10/23 1:40 PM
there is convincing evidence that the alternative hypothesis is away from the null hypothesis value. ˆ ) p proportion. 0 p ) are at least
single best guess for the value of a population parameter.
• Large Counts: Both np and (1n −
• Large Counts: Both ˆ np and (1n − are at least 10.
0
You also learned that a confidence interval, also known as SECTION 6B Confidence Intervals for a That is, the number of successes and the number of Calculate a confidence interval for a population proportion. 6B 546 R2

10, where 0 p is the proportion specified by the null
an interval estimate, provides an interval of plausible values Population Proportion failures in the sample are both at least 10. Construct and interpret a one-sample z interval for a proportion. 6B 548 R2
hypothesis.
for a parameter based on sample data. To interpret a confi- Describe how the sample size and confidence level affect the margin of error. 6B 552 R2
Formula
Inference for a Population Proportion
dence interval, say, “We are C % confident that the interval In this section, you learned how to construct and interpret ˆ(1p − ˆ ) p z = ˆ p − 0 p
ˆ p ±
* z
−
)
0 p
(1 p
from to captures the confidence intervals for a population proportion. Three con- n Determine the sample size required to obtain a confidence interval for a 6B 554 R3
0
Confidence Interval for p
Significance Test for p
population proportion with a specified margin of error.
[parameter in context],” where C is the confidence level of the ditions must be met to ensure that the observations in the sam- n
interval. You can use a confidence interval to evaluate a claim
Name (TI-83/84) One-sample z interval for p (1-PropZInt) ple are independent and that the sampling distribution of ˆ p State appropriate hypotheses for a significance test about a population 6C 564 R4, R5, R6
P-value from the standard normal distribution.
One-sample z test for p (1-PropZTest) * from the standard normal distribution.
Critical value z
about the value of a population parameter. is approximately normal. First, the data used to calculate the parameter.
: 0 H p =
Not applicable.
0 p
Null Hypothesisence level C describes the percentage of interval must come from a random sample from the popula- Interpret a P- value in context. 6C 565 R4
The confid
confidence intervals that we expect to capture the value tion of interest (the Random condition). When the sample is Make an appropriate conclusion for a significance test. 6C 568 R4, R5
Conditions • Random: The data come from a random sample from • Random: The data come from a random sample from Check the conditions for performing a test about a population proportion. 6D 574 R5
selected without replacement from the population, the sample
of the parameter in repeated sampling. To interpret a C %
What Did You Learn?
the population of interest.
the population of interest.
confidence level, say, “If we took many samples of the same size should be less than 10% of the population size (the 10% Calculate the standardized test statistic and P- value for a test about a
10%: When sampling without replacement, n <
10%: When sampling without replacement, n <
0.10N .
Use the WHAT DID YOU LEARN? table to verify your mastery of each topic and to 0.10N . population proportion. Relevant Unit 6D 577 R5
Related Example
Learning Target
• Large Counts: Both ˆ np and (1n − are at least 10. • Large Counts: Both np and (1n − 0 p ) are at least Section on Page(s) Review Exercise(s) 6D 582 R5
ˆ ) p
Perform a one-sample z test for a proportion.
0
find help when needed. All of the individual LEARNING TARGETS are listed with Interpret a Type I error and a Type II error in context and give a consequence 6D 584 R6
533
Interpret a confidence interval in context.
6A
R1, R2
10, where 0 p is the proportion specified by the null
That is, the number of successes and the number of
Use a confidence interval to make a decision about the value of a
of each type of error.
hypothesis.
failures in the sample are both at least 10.
534
R1, R5
6A
references to the sections in which they are introduced, as well as related examples Interpret the power of a significance test and describe which factors affect 6D 590 R6
parameter.
Formula
0 p
6A
ˆ(1 p
07_StarnesTPS7e_40934_un06_529_642.indd 599 p − ˆ ) z = ˆ p − Interpret a confidence level in context. the power of a test. 536 R1
09/10/23 1:40 PM
* z
ˆ p ±
and relevant Unit Review Exercises. And, of course, watch the Unit Review Exercise
−
)
(1 p
0 p
n
0
n
Videos to get tips on solving these multifaceted problems.
Critical value z* from the standard normal distribution. P-value from the standard normal distribution. UNIT 6, PART I REVIEW EXERCISES

These exercises are designed to help you review the important R3 Do you go to church? (6B) The Gallup polling organi-

concepts and skills of the unit. zation plans to ask a random sample of adults whether
07_StarnesTPS7e_40934_un06_529_642.indd 601 09/10/23 1:40 PM
What Did You Learn? R1 Sports fans (6A) Are you a sports fan? That’s the they attended a religious service in the past 7 days. How
large a sample would be required to obtain a margin
question the Gallup polling organization asked a ran- of error of at most 0.01 in a 99% confidence interval

Related Example Relevant Unit dom sample of 1527 U.S. adults. Gallup reported for the population proportion who would say that they
37
Learning Target Section on Page(s) Review Exercise(s) that a 95% confidence interval for the proportion attended a religious service in the past 7 days?
of all U.S. adults who are sports fans is 0.565 to
Interpret a confidence interval in context. 6A 533 R1, R2 0.615. R4 Signature verification (6C) When a petition is submit-

Use a confidence interval to make a decision about the value of a (a) Interpret the confidence interval. ted to government officials to put a political candidate’s

parameter. 6A 534 R1, R5 (b) Interpret the confidence level. name on a ballot, a certain number of valid voters’

signatures are required. Rather than check the valid-

Interpret a confidence level in context. 6A 536 R1 (c) Based on the interval, is there convincing evidence ity of all the signatures, officials often randomly select
that a majority of U.S. adults are sports fans? Explain a sample of signatures for verification and perform a
your answer. significance test to see if the true proportion of signa-
tures is less than the required value. Suppose a petition
Running red lights (6B) A random sample of 880
R2

xxiv U.S. drivers were asked, “Recalling the last 10 traffic has 30,000 signatures and 18,000 valid signatures are
required for a candidate to be on the ballot — which
lights you drove through, how many of them were means at least 60% of the signatures on this petition
red when you entered the intersection?” Of the 880 must be valid. The officials select a random sample of
respondents, 171 admitted that at least one light had 300 signatures and find that 171 are valid. Do these
been red. 38 data provide convincing evidence that the proportion of
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only, all other uses prohibited - Do Not Copy or Post in Any Form.
Construct and interpret a 95% confidence interval for
(a)
all signatures that are valid is less than 0.6?
07_StarnesTPS7e_40934_un06_529_642.indd 601 09/10/23 1:40 PM the population proportion. (a) State the hypotheses we are interested in testing.

(b) Explain two ways you could reduce the margin of (b) The P- value for this test is 0.1444. Interpret this value.

error of this confidence interval. What are the draw-

backs to these actions? (c) What conclusion should you make?
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