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AP STATISTICS EXAM REVIEW by DAVID CUSTER (click on topic of choice) TOPIC I: TOPIC I: Describing Data Describing Data (15 questions) (15 questions) TOPIC.

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Presentation on theme: "AP STATISTICS EXAM REVIEW by DAVID CUSTER (click on topic of choice) TOPIC I: TOPIC I: Describing Data Describing Data (15 questions) (15 questions) TOPIC."— Presentation transcript:

1 AP STATISTICS EXAM REVIEW by DAVID CUSTER (click on topic of choice) TOPIC I: TOPIC I: Describing Data Describing Data (15 questions) (15 questions) TOPIC II: TOPIC II: Experimental Design Experimental Design (15 questions) (15 questions) TOPIC III: TOPIC III: Probability (18 questions) (18 questions) TOPIC IV: TOPIC IV: Inference (15 questions) (15 questions)

2 TOPIC I: Describing Data Univariate DataUnivariate Data Normal DistributionsNormal Distributions Bivariate DataBivariate Data Topic I back to main back to main

3 Which observation has the higher z-score? I. x=25.4; μ =12.9; σ=3.7 II. x=25.4; μ=15.3; σ=2.7 A.I B.II C.z-scores are equal D.cannot be determined since we don’t know the standard deviations of the populations E.cannot be determined since we don’t know if the populations are normal I.1. Topic I Menu

4 Which observation has the higher z-score? I. x=25.4; μ =12.9; σ=3.7 II. x=25.4; μ=15.3; σ=2.7 A.I B.II C.z-scores are equal D.cannot be determined since we don’t know the standard deviations of the populations E.cannot be determined since we don’t know if the populations are normal I.1. SOLUTION The z-score for I is 3.38; the z-score for II is 3.74 Topic I Menu

5 The Quartile and Percentile positions of the value 10 in the set {10, 6, 8, 9, 12, 17, 32, 16} are: A.Q3; 38 B.Q2; 38 C.Q2; 26 D.Q3; 26 E.none of these I.2. Topic I Menu

6 The Quartile and Percentile positions of the value 10 in the set {10, 6, 8, 9, 12, 17, 32, 16} are: A.Q3; 38 B.Q2; 38 C.Q2; 26 D.Q3; 26 E.none of these I.2. SOLUTION When the values are in order, there are 3 values below 10. This puts 10 in the 37.5 percentile and the 2 nd quartile. Topic I Menu

7 The Standard Deviation of the set {5, 7, 7, 8, 10, 11} is 2. Which of the following sets also has a Standard Deviation of 2? A.{4, 5, 8, 12, 14} B.{2, 4, 6, 8, 10, 12} C.{3, 5, 5, 6, 8, 9} D.{10, 14, 14, 16, 20, 22} E.none of the above I.3. Topic I Menu

8 The Standard Deviation of the set {5, 7, 7, 8, 10, 11} is 2. Which of the following sets also has a Standard Deviation of 2? A.{4, 5, 8, 12, 14} B.{2, 4, 6, 8, 10, 12} C.{3, 5, 5, 6, 8, 9} D.{10, 14, 14, 16, 20, 22} E.none of the above I.3. There is a uniform decrease of 2 units in this set. The st. deviation must be the same. SOLUTION Topic I Menu

9 If 12% of the values of a data set lie between a and b and d is added to each value, then which of the following is true? A.12% still lies between a and b B.12% lies between a + d and b + d C.(12+d)% lies between a and b D.(12+d)% lies between a + d and b + d E.there is no way to tell how much data is between a and b I.4. Topic I Menu

10 If 12% of the values of a data set lie between a and b and d is added to each value, then which of the following is true? A.12% still lies between a and b B.12% lies between a + d and b + d C.(12+d)% lies between a and b D.(12+d)% lies between a + d and b + d E.there is no way to tell how much data is between a and b I.4. A uniform shift of all the data maintains the percentages of data in shifted intervals. SOLUTION Topic I Menu

11 If a distribution is relatively symmetric and mount-shaped, order the following (from least to greatest) 1. a z-score of 1 2. the value of Q3 3. a value in the 70 th percentile A.1, 2, 3 B.1, 3, 2 C.3, 2, 1 D.3, 1, 2 E.2, 3, 1 I.5. Topic I Menu

12 If a distribution is relatively symmetric and mount-shaped, order the following (from least to greatest) 1. a z-score of 1 2. the value of Q3 3. a value in the 70 th percentile A.1, 2, 3 B.1, 3, 2 C.3, 2, 1 D.3, 1, 2 E.2, 3, 1 I.5. SOLUTION The percentile of a z-score of 1 is about 84%, and the percentile of Q3 is 75% Topic I Menu

13 Which of the following would NOT be a correct interpretation of a correlation coefficient of r = -.30 A.The variables are inversely related B.The coefficient of determination is 0.09 C.30% of the variation between the variables is linear D.There exists a weak relationship between the variables E.All are correct I.6. Topic I Menu

14 Which of the following would NOT be a correct interpretation of a correlation coefficient of r = -.30 A.The variables are inversely related B.The coefficient of determination is 0.09 C.30% of the variation between the variables is linear D.There exists a weak relationship between the variables E.All are correct I.6. SOLUTION The value of r 2 explains the variation between the variables. Not r. Topic I Menu

15 Which of the following displays is best suited for categorical data? A.Box Plot B.Bar Graph C.Stem and Leaf Plot D.Dot Plot E.Scatterplot I.7. Topic I Menu

16 Which of the following displays is best suited for categorical data? A.Box Plot B.Bar Graph C.Stem and Leaf Plot D.Dot Plot E.Scatterplot I.7. SOLUTION In a bar graph, each column is separate, allowing for categorical separation. Topic I Menu

17 Linear regression usually employs the method of least squares. Which of the following is the quantity that is minimized by the least squares process? A. A. B. B. C. C. D. D. E. E. I.8. Topic I Menu

18 Linear regression usually employs the method of least squares. Which of the following is the quantity that is minimized by the least squares process? A. A. B. B. C. C. D. D. E. E. I.8. SOLUTION Least Squares Regression minimizes the residuals in the y-direction. Topic I Menu

19 Which of the following is NOT true? A.Two sets of data can have the same means but different variances B.Two sets of data can have the same variance but different means C.Two different values in a data set can have the same z-score D.All the absolute values of z-scores for a data set can be equal E.All of the above are true I.9. Topic I Menu

20 Which of the following is NOT true? A.Two sets of data can have the same means but different variances B.Two sets of data can have the same variance but different means C.Two different values in a data set can have the same z-score D.All the absolute values of z-scores for a data set can be equal E.All of the above are true I.9. SOLUTION Since each value is a distinct distance from the mean, the z-scores must all be different Topic I Menu

21 In a symmetric, mount-shaped distribution, what percentile has a z-score of -2? I.10. Topic I Menu

22 Topic I Menu In a symmetric, mount-shaped distribution, what percentile has a z-score of -2? I.10. SOLUTION We should be able to approximate this with the normal distribution. Area to the left of -2? On the TI-83: Normalcdf(-1E99, -2) = th percentile

23 Lee’s z-score on his math test was 1.5. The class average was a 62.1 and the variance was What was Lee’s actual grade on the test? A.60 B.62 C.64 D.66 E.68 I.11. Topic I Menu

24 Lee’s z-score on his math test was 1.5. The class average was a 62.1 and the variance was What was Lee’s actual grade on the test? A.60 B.62 C.64 D.66 E.68 I.11. SOLUTION We need to solve the following equation for x: so x=66 Topic I Menu

25 Does the following problem have a unique solution? If so, find it. If not, show at least two answers: 5 numbers have Q1=12, Median=15, Q3=18. Find the mean. I.12. Topic I Menu

26 Does the following problem have a unique solution? If so, find it. If not, show at least two answers: 5 numbers have Q1=12, Median=15, Q3=18. Find the mean. I.12. SOLUTION {11, 13, 15, 17, 19} μ=15 { 9, 15, 15, 17, 19} μ=15 the data sets are not unique, but the mean is always 15! YES!!! surprisingly! Topic I Menu

27 The average grade on a math test given to two sections is Section I has 27 students with a mean grade of If the mean grade of Section 2 is 65.30, how many students are in section 2? I.13. Topic I Menu

28 Topic I Menu The average grade on a math test given to two sections is Section I has 27 students with a mean grade of If the mean grade of Section 2 is 65.30, how many students are in section 2? I.13. SOLUTION 23 students. we arrive at n=23

29 If the mean of 75 values is 52.6 and the mean of 25 values is 48.4; find the mean of all 100 values. A B C D E I.14. Topic I Menu

30 If the mean of 75 values is 52.6 and the mean of 25 values is 48.4; find the mean of all 100 values. A B C D E I.14. SOLUTION 75(52.6) + 25(48.4) 100 Topic I Menu

31 In skewed-right distributions, what is most frequently the relationship of the mean, median, and mode? A.mean > median > mode B.median > mean > mode C.mode > median > mean D.mode > mean > median E.mean > mode > median I.15. Topic I Menu

32 Topic I Menu In skewed-right distributions, what is most frequently the relationship of the mean, median, and mode? A.mean > median > mode B.median > mean > mode C.mode > median > mean D.mode > mean > median E.mean > mode > median I.15. SOLUTION The median is resistant, the mean, not at all. So a right skewed distribution will have a mean much higher than median, much higher than mode.

33 TOPIC II: Experimental Design SamplingSampling Designing ExperimentsDesigning Experiments Observational StudiesObservational Studies Topic II back to main back to main

34 A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. The study is categorized as: A.a census B.a survey sample C.an observational study D.a designed experiment E.none of these II.1. Topic II Menu

35 A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. The study is categorized as: A.a census B.a survey sample C.an observational study D.a designed experiment E.none of these II.1. SOLUTION The director is observing behavior, not implementing treatments on the group Topic II Menu

36 A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. If the director only looks at those in one department, she is performing: A.a simple random sample B.a quota sample C.a convenience sample D.a multi-stage cluster sample E.a census II.2. Topic II Menu

37 Topic II Menu SOLUTION Without a properly randomized selection, she is introducing bias to the study. A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. If the director only looks at those in one department, she is performing: A.a simple random sample B.a quota sample C.a convenience sample D.a multi-stage cluster sample E.a census II.2.

38 A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. If the director selects 50 employees at random and categorizes by gender, she is: A.blocking for gender B.testing for a lurking variable C.promoting sexual harassment D.testing for bias E.none of these II.3. Topic II Menu

39 A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. If the director selects 50 employees at random and categorizes by gender, she is: A.blocking for gender B.testing for a lurking variable C.promoting sexual harassment D.testing for bias E.none of these II.3. SOLUTION She is investigating whether gender affects lunchtime behavior Topic II Menu

40 Which of the following is NOT a concern in data collection? A.lurking variables B.blocking C.bias D.non-response E.all of the above are concerns II.4. Topic II Menu

41 Which of the following is NOT a concern in data collection? A.lurking variables B.blocking C.bias D.non-response E.all of the above are concerns II.4. SOLUTION Even blocking methods need to be analyzed Topic II Menu

42 Which of the following is NOT a valid sample design? A.Code every member of a population and select 100 randomly chosen members B.Divide a population by gender and select 50 individuals randomly from each group C.Select individuals randomly and place into gender groups until you have the same proportion as in the population D.Select five homerooms at random from all the homerooms in a large high school. E.All of these are valid II. 5. Topic II Menu

43 Which of the following is NOT a valid sample design? A.Code every member of a population and select 100 randomly chosen members B.Divide a population by gender and select 50 individuals randomly from each group C.Select individuals randomly and place into gender groups until you have the same proportion as in the population D.Select five homerooms at random from all the homerooms in a large high school. E.All of these are valid II. 5. SOLUTION But you may be able to question the validity of answer (C) Topic II Menu

44 An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. The variable of interest is: A.the doctor’s specialty, e.g. cardiology, obstetrics, etc. B.the number of doctors who are cardiologists C.all cardiologists in the American Medical Association directory D.a random sample of 100 cardiologists E.none of these II.6. Topic II Menu

45 An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. The variable of interest is: A.the doctor’s specialty, e.g. cardiology, obstetrics, etc. B.the number of doctors who are cardiologists C.all cardiologists in the American Medical Association directory D.a random sample of 100 cardiologists E.none of these II.6. SOLUTION we are interested in the percentage of doctors who have been sued Topic II Menu

46 An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. The population of interest is: A.the set of all doctors who were sued for malpractice B.the set of cardiologists who were sued for malpractice C.all doctors D.all cardiologists E.all doctors who have malpractice insurance II.7. Topic II Menu

47 An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. The population of interest is: A.the set of all doctors who were sued for malpractice B.the set of cardiologists who were sued for malpractice C.all doctors D.all cardiologists E.all doctors who have malpractice insurance II.7. SOLUTION Just cardiologists, not all doctors. Topic II Menu

48 An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. Which could be used to gather the data? A.a designed experiment B.a census of all cardiologists C.an observational study of randomly selected cardiologists D.a survey sent to randomly selected cardiologists E.any answer except (A) II.8. Topic II Menu

49 Topic II Menu An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. Which could be used to gather the data? A.a designed experiment B.a census of all cardiologists C.an observational study of randomly selected cardiologists D.a survey sent to randomly selected cardiologists E.any answer except (A) II.8. SOLUTION This population of this observation al study is too large to track everyone down.

50 Which of the following is NOT a source of bias in sample surveys? A.non-response B.wording of questions C.voluntary response D.use of a telephone survey E.all are sources of bias II.9. Topic II Menu

51 Which of the following is NOT a source of bias in sample surveys? A.non-response B.wording of questions C.voluntary response D.use of a telephone survey E.all are sources of bias II.9. SOLUTION and don’t forget… even a huge sample size can’t correct a poorly selected sample. Remember the Literary Digest Poll! Topic II Menu

52 Which of the following is NOT a requirement of a controlled experiment? A.control B.comparison C.replication D.randomization E.all of these are required II.10. Topic II Menu

53 Which of the following is NOT a requirement of a controlled experiment? A.control B.comparison C.replication D.randomization E.all of these are required II.10. SOLUTION Topic II Menu

54 A randomized block design is NOT: A.similar to a stratified random sample for surveys B.a strategy to control for an influence that would affect the outcome of the experiment C.a strategy that depends on randomization D.only used for gender comparisons E.all of these describe a randomized block design. II.11. Topic II Menu

55 A randomized block design is NOT: A.similar to a stratified random sample for surveys B.a strategy to control for an influence that would affect the outcome of the experiment C.a strategy that depends on randomization D.only used for gender comparisons E.all of these describe a randomized block design. II.11. SOLUTION You can block with any categorical variables! Topic II Menu

56 A research team is comparing performance in AP Statistics based on whether traditional or activity-based instruction methods were used. The final grades of 500 students will be collected. The population of interest is: A.the 500 students chosen B.the students taught by activity-based statistics C.the students taught by traditional methods D.all students in high school. E.none of these II.12. Topic II Menu

57 Topic II Menu A research team is comparing performance in AP Statistics based on whether traditional or activity-based instruction methods were used. The final grades of 500 students will be collected. The population of interest is: A.the 500 students chosen B.the students taught by activity-based statistics C.the students taught by traditional methods D.all students in high school. E.none of these II.12. SOLUTION We are interested in knowing about ALL AP STATISTICS STUDENTS

58 A research team is comparing performance in AP Statistics based on whether traditional or activity-based instruction methods were used. The final grades of 500 students will be collected. An appropriate design for the study is: A.a blocked design experiment B.a stratified random sample C.a completely randomized design D.a simple random sample E.none of these II.13. Topic II Menu

59 A research team is comparing performance in AP Statistics based on whether traditional or activity-based instruction methods were used. The final grades of 500 students will be collected. An appropriate design for the study is: A.a blocked design experiment B.a stratified random sample C.a completely randomized design D.a simple random sample E.none of these II.13. SOLUTION B or C. Either one is fine. Topic II Menu

60 A survey is to be conducted in your school. There is to be a total of 40 students in the sample. Describe how you would choose the participants if there are to be the same number of freshmen, sophomores, juniors, and seniors in the sample. II.14. Topic II Menu

61 A survey is to be conducted in your school. There is to be a total of 40 students in the sample. Describe how you would choose the participants if there are to be the same number of freshmen, sophomores, juniors, and seniors in the sample. II.14. SOLUTION Select a simple random sample of 10 from each class. Topic II Menu

62 II.15. A survey is to be conducted in your school. There is to be a total of 40 students in the sample. Describe how you would choose the participants if there are to be the same number of males and females in the sample Topic II Menu

63 II.15. A survey is to be conducted in your school. There is to be a total of 40 students in the sample. Describe how you would choose the participants if there are to be the same number of males and females in the sample SOLUTION Select a simple random sample of 20 males and 20 females. Topic II Menu

64 TOPIC III: Probability Random VariablesRandom Variables Binomial DistributionsBinomial Distributions Geometric DistributionsGeometric Distributions Sampling DistributionsSampling Distributions Topic III back to main back to main

65 If 3 people, Joe, Betsy, and Sue, play a game in which Joe has a 25% chance of winning and Betsy has a 40% chance of winning, what is the probability that Sue will win? A.25% B.35% C.40% D.65% E.cannot be determined III.1. Topic III Menu

66 If 3 people, Joe, Betsy, and Sue, play a game in which Joe has a 25% chance of winning and Betsy has a 40% chance of winning, what is the probability that Sue will win? A.25% B.35% C.40% D.65% E.cannot be determined III.1. SOLUTION Assuming one winner, the probabilities must add up to 100% Topic III Menu

67 A local law enforcement agency published the following chart. The percentage of altercations involving at least one teenager is: A.8% B.37% C.45% D.55% E.82% III.2. Altercations Between Percent Two teens 45% A teen and an adult 37% Two adults 18% Topic III Menu

68 A local law enforcement agency published the following chart. The percentage of altercations involving at least one teenager is: A.8% B.37% C.45% D.55% E.82% III.2. Altercations Between Percent Two teens 45% A teen and an adult 37% Two adults 18% SOLUTION p(two teens) + p(teen and adult) = Topic III Menu

69 What proportion of Republicans are over 50? A.61/238 B.32/96 C.96/238 D.32/61 E.cannot be determined III.3. DemocratRepublicanIndepend 18 – – – over Topic III Menu

70 What proportion of Republicans are over 50? A.61/238 B.32/96 C.96/238 D.32/61 E.cannot be determined III.3.DemocratRepublicanIndepend 18 – – – over SOLUTION There are 96 Republicans of whom 32 are over age 50 Topic III Menu

71 If one person is chosen at random, what is the probability he is a Democrat between 41 and 50 years old? A.17/238 B.17/88 C.61/238 D.17/61 E.88/238 III.4. DemocratRepublicanIndepend 18 – – – over Topic III Menu

72 If one person is chosen at random, what is the probability he is a Democrat between 41 and 50 years old? A.17/238 B.17/88 C.61/238 D.17/61 E.88/238 III.4.DemocratRepublicanIndepend 18 – – – over SOLUTION Total # of adults is are democrats between 41 and 50 Topic III Menu

73 Given that a person chosen is between 31 and 40, what is the probability the person is an Independent? A.10/238 B.10/63 C.10/54 D.54/238 E.63/238 III.5. DemocratRepublicanIndepend 18 – – – over Topic III Menu

74 Given that a person chosen is between 31 and 40, what is the probability the person is an Independent? A.10/238 B.10/63 C.10/54 D.54/238 E.63/238 III.5.DemocratRepublicanIndepend 18 – – – over SOLUTION There are 63 people between 31 and 40 of whom 10 are Independent. Topic III Menu

75 What proportion of the citizens sampled are over 50 OR Independent? A.54/238 B.61/238 C.100/238 D.115/238 E.cannot be determined III.6. DemocratRepublicanIndepend 18 – – – over Topic III Menu

76 What proportion of the citizens sampled are over 50 OR Independent? A.54/238 B.61/238 C.100/238 D.115/238 E.cannot be determined III.6.DemocratRepublicanIndepend 18 – – – over SOLUTION 61/ /238 – 15/ adults are in both categories Topic III Menu

77 If P(A)=0.4, P(B)=0.2, and P(A and B)= 0.08 Which is true? A.Events A and B are independent and mutually exclusive B.Events A and B are independent but not mutually exclusive C.Events A and B are mutually exclusive but not independent D.Events A and B are neither independent nor mutually exclusive E.Events A and B are independent but whether they are mutually exclusive cannot be determined. III.7. Topic III Menu

78 If P(A)=0.4, P(B)=0.2, and P(A and B)= 0.08 Which is true? A.Events A and B are independent and mutually exclusive B.Events A and B are independent but not mutually exclusive C.Events A and B are mutually exclusive but not independent D.Events A and B are neither independent nor mutually exclusive E.Events A and B are independent but whether they are mutually exclusive cannot be determined. III.7. SOLUTION --Since P(A and B) is non-zero, they are not mutually exclusive. --Since P(A and B) does not equal P(A)P(B), the events are independent—they just happen in sequence Topic III Menu

79 What is the probability that a family with 6 children will have 3 boys and 3 girls? POSSIBLE NAME OF DISTRIBUTION? 1.Binomial Model 2.Geometric Model 3.Uniform Model 4.Normal Model III.8. Topic III Menu

80 What is the probability that a family with 6 children will have 3 boys and 3 girls? POSSIBLE NAME OF DISTRIBUTION? 1.Binomial Model 2.Geometric Model 3.Uniform Model 4.Normal Model III.8. SOLUTION This is a binomial distribution: n=6 p=0.5 6 C 3 (0.5) 3 (0.5) 3 = x=3 Topic III Menu

81 What is the probability that a person is over 6 feet tall if the mean height of her age group is 5’6” and a standard deviation of 10”? POSSIBLE NAME OF DISTRIBUTION? 1.Binomial Model 2.Geometric Model 3.Uniform Model 4.Normal Model III.9. Topic III Menu

82 What is the probability that a person is over 6 feet tall if the mean height of her age group is 5’6” and a standard deviation of 10”? POSSIBLE NAME OF DISTRIBUTION? 1.Binomial Model 2.Geometric Model 3.Uniform Model 4.Normal Model III.9. SOLUTION This is a normal distribution: normalcdf(6, 1E99, 5+6/12, 10/12) = Topic III Menu

83 What is the probability that a shipment of 100 fruit will have no more than 6 rotten fruits if the probability that any one fruit is rotten is 0.04? POSSIBLE NAME OF DISTRIBUTION? 1.Binomial Model 2.Geometric Model 3.Uniform Model 4.Normal Model III.10. Topic III Menu

84 What is the probability that a shipment of 100 fruit will have no more than 6 rotten fruits if the probability that any one fruit is rotten is 0.04? POSSIBLE NAME OF DISTRIBUTION? 1.Binomial Model 2.Geometric Model 3.Uniform Model 4.Normal Model III.10. SOLUTION Did you think to use the normal approximation? μ = np = 4 σ =√p(1–p)/n =.0196 normalcdf(-1E99, 6, 4,.0196) = 1.00 How come that didn’t work? This is a binomial distribution: binomcdf(100, 0.04, 6) = Topic III Menu

85 What is the probability that the first base hit will occur during the fourth at-bat if the probability that the hitter gets a base hit is 0.27 for any at-bat? POSSIBLE NAME OF DISTRIBUTION? 1.Binomial Model 2.Geometric Model 3.Uniform Model 4.Normal Model III.11. Topic III Menu

86 What is the probability that the first base hit will occur during the fourth at-bat if the probability that the hitter gets a base hit is 0.27 for any at-bat? POSSIBLE NAME OF DISTRIBUTION? 1.Binomial Model 2.Geometric Model 3.Uniform Model 4.Normal Model III.11. SOLUTION This is a geometric (waiting time) distribution: n=??? (there isn’t one! That’s why it’s Geometric!) p=0.27; x=4 geometpdf(0.27, 4) = (.73)(.73)(.73)(.27) = Topic III Menu

87 Suppose a basketball player scores 70% of her free throws. Assume each shot is independent and the probability is the same on each trial. Find the probability she scores on 3 of her next 5 attempts III.12. Topic III Menu

88 Suppose a basketball player scores 70% of her free throws. Assume each shot is independent and the probability is the same on each trial. Find the probability she scores on 3 of her next 5 attempts III.12. SOLUTION This is a binomial distribution: n=5 p= C 3 (0.7) 3 (0.3) 2 = x=3 Topic III Menu

89 Suppose a basketball player scores 70% of her free throws. Assume each shot is independent and the probability is the same on each trial. Find the probability that the first time she scores is on her 3 rd attempt III.13. Topic III Menu

90 Suppose a basketball player scores 70% of her free throws. Assume each shot is independent and the probability is the same on each trial. Find the probability that the first time she scores is on her 3 rd attempt III.13. SOLUTION This is a geometric distribution: n=??? (we wait ‘til a success—there’s no n!) p=0.70; x=3 geometpdf(0.7, 3) = (.3)(.3)(.7) = Topic III Menu

91 The Central Limit Theorem for sample means is critical because… A.It states that for large sample sizes, the population distribution is approximately normal B.It states that for large sample sizes, the sample is approximately normal C.It states that for any population, the sampling distribution is normal regardless of sample size D.It states that for large sample sizes, the sampling distribution is approximately normal regardless of the population distribution E.It states that for any sample size, the sampling distribution is normal III.14. Topic III Menu

92 The Central Limit Theorem for sample means is critical because… A.It states that for large sample sizes, the population distribution is approximately normal B.It states that for large sample sizes, the sample is approximately normal C.It states that for any population, the sampling distribution is normal regardless of sample size D.It states that for large sample sizes, the sampling distribution is approximately normal regardless of the population distribution E.It states that for any sample size, the sampling distribution is normal III.14. SOLUTION Topic III Menu

93 The amount of time it takes a high school class of 1000 freshmen to swim 10 lengths of the school pool has a distribution that is skewed left due to some excellent swimmers. The mean amount of time needed is 9.2 minutes and the standard deviation is 5.3 min. If 64 students are chosen at random, then what is the probability their mean time will exceed 10 minutes? III.15. Topic III Menu

94 The amount of time it takes a high school class of 1000 freshmen to swim 10 lengths of the school pool has a distribution that is skewed left due to some excellent swimmers. The mean amount of time needed is 9.2 minutes and the standard deviation is 5.3 min. If 64 students are chosen at random, then what is the probability their mean time will exceed 10 minutes? III.15. SOLUTION normalcdf(10, 1E99, 9.2, 5.3/√64) = Topic III Menu

95 It is assumed that 50% of all people catch one or more colds each year. What is the probability that out of 400 randomly selected people, 216 or more will catch one or more colds this year? A B C D.0.11 E.0.55 III.16. Topic III Menu

96 It is assumed that 50% of all people catch one or more colds each year. What is the probability that out of 400 randomly selected people, 216 or more will catch one or more colds this year? A B C D.0.11 E.0.55 III.16. SOLUTION We want p(p > 216/400 =.054) Since np and n(1-p) exceed 10, we can apply the normal approximation with µ=.5 and σ =.025 Normalcdf(.54, 1E99,.5,.025) =.055 Topic III Menu

97 The decision whether the distribution ofo a sample mean follows a normal or a t- distribution depends on: A.Sample size B.Whether you have the actual data or only statistics of the data C.Whether you know the population standard deviation D.Whether np>10 and n(1-p)>10 E.None of the above III.17. Topic III Menu

98 The decision whether the distribution ofo a sample mean follows a normal or a t- distribution depends on: A.Sample size B.Whether you have the actual data or only statistics of the data C.Whether you know the population standard deviation D.Whether np>10 and n(1-p)>10 E.None of the above III.17. SOLUTION Topic III Menu

99 National studies show that 14% of male teenagers and 12% of female teenagers will be involved in a major traffic accident while driving. What’s the probability that independent samples of 100 female teens and 75 male teens will have results that differ by more than 3% in either direction? A..042 B..085 C..42 D..85 E.Cannot be determined III.18. Topic III Menu

100 National studies show that 14% of male teenagers and 12% of female teenagers will be involved in a major traffic accident while driving. What’s the probability that independent samples of 100 female teens and 75 male teens will have results that differ by more than 3% in either direction? A..042 B..085 C..42 D..85 E.Cannot be determined III.18. SOLUTION For both genders, np and n(1-p) exceed 5 so we can use the CLT. z = ( ) =.1938 √(.14*.86/ *.88/100) Using normalcdf(.1938, 1E99) =.423 But we want both directions so add in the other tail to get an answer of.846

101 TOPIC IV: Inference Inference on MeansInference on Means Inference on ProportionsInference on Proportions Inference on 2-way TablesInference on 2-way Tables Inference on RegressionInference on Regression Topic IV back to main back to main

102 Given H0: μ=30, HA: μ<30, if you conclude that the mean is less than 30 when it is actually 27… A.you have made a type II error B.you have made a type I error C.the result of your test was not significant D.you have drawn a correct conclusion E.all of the above are true IV.1. Topic IV Menu

103 Given H0: μ=30, HA: μ<30, if you conclude that the mean is less than 30 when it is actually 27… A.you have made a type II error B.you have made a type I error C.the result of your test was not significant D.you have drawn a correct conclusion E.all of the above are true IV.1. SOLUTION You are rejecting a false null hypothesis. No problems. Correct decision. Topic IV Menu

104 The local news station reports that the 97% confidence interval for a candidate’s support was (43%, 48%). What does the phrase “97% confidence” mean? A.97% of the voters support the candidate B.97% of the time, this candidate’s level of support will be between 43% and 48% C.There is a 97% probability that the true level of support is between 43% and 48% D.There is a 97% probability that any other sample percentage is in the interval (43%, 48%) E.none of these is true IV.2. Topic IV Menu

105 The local news station reports that the 97% confidence interval for a candidate’s support was (43%, 48%). What does the phrase “97% confidence” mean? A.97% of the voters support the candidate B.97% of the time, this candidate’s level of support will be between 43% and 48% C.There is a 97% probability that the true level of support is between 43% and 48% D.There is a 97% probability that any other sample percentage is in the interval (43%, 48%) E.none of these is true IV.2. SOLUTION not (D) because you’re predicting the range of the TRUE proportion; you’re not interested in predicting the range of other samples!!! Correct interpretation: “If this process were repeated over and over again, about 97% of our INTERVALS would capture the true proportion.” Topic IV Menu

106 A study of 20 teachers in a school district indicated that the 95% confidence interval for the mean salary of all teachers is ($38,945, $41,245). What assumptions must be true for this interval to be valid? A.no assumptions are necessary. The CLT applies B.The sample is randomly selected from a population of salaries that is a t-distribution C.the distribution of the sample means is approximately normal D.the distribution of all teachers’ salaries is approximately normal E.None of the above. IV.3. Topic IV Menu

107 A study of 20 teachers in a school district indicated that the 95% confidence interval for the mean salary of all teachers is ($38,945, $41,245). What assumptions must be true for this interval to be valid? A.no assumptions are necessary. The CLT applies B.The sample is randomly selected from a population of salaries that is a t-distribution C.the distribution of the sample means is approximately normal D.the distribution of all teachers’ salaries is approximately normal E.None of the above. IV.3. SOLUTION Because our sample size is only 20, you have to know the population was normal to begin with. (D) is the best response. …I might accept choice (C) too since the CLT is all about when x-bar is normal, which is the goal I suppose. Topic IV Menu

108 In order to reduce the width of a confidence interval, we can: A.increase sample size only B.increase confidence level only C.increase sample size and increase confidence level D.increase sample size and decrease confidence level E.none of these would reduce the width of the interval IV. 4. Topic IV Menu

109 In order to reduce the width of a confidence interval, we can: A.increase sample size only B.increase confidence level only C.increase sample size and increase confidence level D.increase sample size and decrease confidence level E.none of these would reduce the width of the interval IV. 4. SOLUTION Less confidence affords you to predict a much tighter range of values. Topic IV Menu

110 If the 95% confidence interval for μ is (6,9), what conclusion can we draw if we test Ho: μ=10 vs. Ha: μ≠10 at α=.05? A.reject Ho B.fail to reject Ho C.accept Ho D.accept Ha E.There is insufficient information given to draw a conclusion IV.5. Topic IV Menu

111 If the 95% confidence interval for μ is (6,9), what conclusion can we draw if we test Ho: μ=10 vs. Ha: μ≠10 at α=.05? A.reject Ho B.fail to reject Ho C.accept Ho D.accept Ha E.There is insufficient information given to draw a conclusion IV.5. SOLUTION α=.05 A two tailed test of α=.05 is equivalent to a 95% confidence interval. 10 is not within the interval. Topic IV Menu

112 A bakery determines that it will be profitable if the time it takes to decorate a cake does not exceed 45 minutes. The owner documents the time spent on 20 cakes and performs a test. If the P-value of the test is 0.032, then he should conclude: A.at α=.05, fail to reject Ho B.at α=.05, reject Ho C.at α=.03, reject Ho D.at α=.025, reject Ho E.We cannot draw a conclusion from this information IV.6. Topic IV Menu

113 A bakery determines that it will be profitable if the time it takes to decorate a cake does not exceed 45 minutes. The owner documents the time spent on 20 cakes and performs a test. If the P-value of the test is 0.032, then he should conclude: A.at α=.05, fail to reject Ho B.at α=.05, reject Ho C.at α=.03, reject Ho D.at α=.025, reject Ho E.We cannot draw a conclusion from this information IV.6. SOLUTION a p-value of.032 is significant at the 5% level Topic IV Menu

114 Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. What design seems to be employed? A.Matched Pairs Design B.Simple Random Design C.Multi-State Cluster Design D.Independent Samples Design E.Randomized Block Design IV.7. boysgirls n Mean St.Deviation Topic IV Menu

115 Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. What design seems to be employed? A.Matched Pairs Design B.Simple Random Design C.Multi-State Cluster Design D.Independent Samples Design E.Randomized Block Design IV.7.boysgirlsn Mean St.Deviation SOLUTION We have 2 independent samples here: boys and girls. Topic IV Menu

116 Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. What are Ho and Ha to determine if boys’ scores are higher than girls? A.Ho: μ b – μ g = 0; Ha: μ b – μ g < 0 B.Ho: μ b – μ g = 0; Ha: μ b – μ g ≠ 0 C.Ho: μ b – μ g = 0; Ha: μ b – μ g > 0 D.Ho: μ b – μ g < 0; Ha: μ b – μ g = 0 E.Ho: μ b = μ g ; Ha: μ b ≠ μ g IV.8. boysgirls n Mean St.Dev Topic IV Menu

117 Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. What are Ho and Ha to determine if boys’ scores are higher than girls? A.Ho: μ b – μ g = 0; Ha: μ b – μ g < 0 B.Ho: μ b – μ g = 0; Ha: μ b – μ g ≠ 0 C.Ho: μ b – μ g = 0; Ha: μ b – μ g > 0 D.Ho: μ b – μ g < 0; Ha: μ b – μ g = 0 E.Ho: μ b = μ g ; Ha: μ b ≠ μ g IV.8.boysgirlsn Mean St.Dev SOLUTION Another way to write this would be Ho: μ b = μ g ; Ha: μ b > μ g Topic IV Menu

118 Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. Suppose the p-value is We can then conclude: IV.9. boysgirls n Mean St.Dev A.at α=.025, reject Ho B.at α=.02, reject Ho C.at α=.01, reject Ho D.at α=.025, fail to reject Ho E.We cannot draw a conclusion from this information Topic IV Menu

119 Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. Suppose the p-value is We can then conclude: IV.9.boysgirlsn Mean St.Dev A.at α=.025, reject Ho B.at α=.02, reject Ho C.at α=.01, reject Ho D.at α=.025, fail to reject Ho E.We cannot draw a conclusion from this information SOLUTION.0344 is not significant at.025,.02, or.01. We must fail to reject it at these levels. Topic IV Menu

120 The rejection region for a test Ho: p=.4 vs. Ha: p<.4, with n=50 and α=.05 is given by: A.Reject Ho if z > 1.96 or z 1.96 or z < B.Reject Ho if z > or z or z < C.Reject Ho if z < D.Reject Ho if z < E.Reject Ho if t > 2.59 IV.10. Topic IV Menu

121 The rejection region for a test Ho: p=.4 vs. Ha: p<.4, with n=50 and α=.05 is given by: A.Reject Ho if z > 1.96 or z 1.96 or z < B.Reject Ho if z > or z or z < C.Reject Ho if z < D.Reject Ho if z < E.Reject Ho if t > 2.59 IV.10. SOLUTION invnorm(.05) because the entire.05 rejection region is in the left tail. Topic IV Menu

122 The power of a test is described by all of the following EXCEPT A.Power = p( rejecting Ho when Ha is true) B.Power = 1 – β C.Power = α + β D.The calculation of power requires knowing the values of μ0, μa, σ, and α E.All of these are correct descriptions of the concept of power. IV.11. Topic IV Menu

123 The power of a test is described by all of the following EXCEPT A.Power = p( rejecting Ho when Ha is true) B.Power = 1 – β C.Power = α + β D.The calculation of power requires knowing the values of μ0, μa, σ, and α E.All of these are correct descriptions of the concept of power. IV.11. SOLUTION But it IS true that as one increases, the other has to decrease. …just not always by the same amount Topic IV Menu

124 In a matched pairs test of 75 pairs, which of the following assumptions is necessary? A.The distribution of the paired differences should be approximately normal B.The population variances should be equal C.The samples are randomly and independently selected D.The sets of values for each variable are approximately normal E.None of these assumptions is necessary. IV.12. Topic IV Menu

125 In a matched pairs test of 75 pairs, which of the following assumptions is necessary? A.The distribution of the paired differences should be approximately normal B.The population variances should be equal C.The samples are randomly and independently selected D.The sets of values for each variable are approximately normal E.None of these assumptions is necessary. IV.12. SOLUTION Topic IV Menu And being good stats students, you will either make sure n>25 or do a quick little graph to show there aren’t any outliers. Right?

126 A pharmaceutical company claims that 50% of adult males living in a city in the Midwest get at least two colds per year. A random sample of 100 adult males living in the city reported that only 42% got two or more colds. Do these data provide evidence (at the 5% significance level) that the true proportion of people is less than 50% IV.13. Topic IV Menu

127 A pharmaceutical company claims that 50% of adult males living in a city in the Midwest get at least two colds per year. A random sample of 100 adult males living in the city reported that only 42% got two or more colds. Do these data provide evidence (at the 5% significance level) that the true proportion of people is less than 50% Of course you would want to run through the entire test properly, but… we fail to reject at the 5% level. there is not enough evidence to say the proportion is less than 50% IV.13. SOLUTION Topic IV Menu

128 If a 90% confidence interval for the slope of a regression line does not contain 0, then which of the following is a valid conclusion? A.The confidence interval is not valid B.A significance test will not be significant at the 10% level C.There is sufficient evidence to conclude that the slope of the true regression line is 0 D.There is sufficient evidence to conclude that the slope of the true regression line is not 0. E.None of these is valid. IV.14. Topic IV Menu

129 If a 90% confidence interval for the slope of a regression line does not contain 0, then which of the following is a valid conclusion? A.The confidence interval is not valid B.A significance test will not be significant at the 10% level C.There is sufficient evidence to conclude that the slope of the true regression line is 0 D.There is sufficient evidence to conclude that the slope of the true regression line is not 0. E.None of these is valid. IV.14. SOLUTION If 0 isn’t in the interval, we are 90% confident that the slope is NOT zero. This is exactly what a regression test would conclude: the slope is NOT zero. Topic IV Menu

130 Topic IV Menu What is the expected number of males who prefer chocolate? A.27.8 B.29.2 C.31.3 D.36.3 E.None of these IV.15.malefemalechocolate3216 vanilla144 strawberry310

131 What is the expected number of males who prefer chocolate? A.27.8 B.29.2 C.31.3 D.36.3 E.None of these IV.15.malefemalechocolate3216 vanilla144 strawberry310 SOLUTION 48 of 79 people prefer chocolate (60.76%), so we expect 60.76% of the 49 males to like chocolate..6076(49) = or if you prefer the book formula: (row total x column total) total number = Topic IV Menu


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