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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
                   © 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy
               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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