Today is Wednesday, January 8th

Notes: Paired T-tests Pages 144-146

Group Quiz Tomorrow

Homework Notification Assigned: CWK 5.6 Paired T-test

Due Thursday, January 9th

Death Packet: Means & Errors

Due Wednesday, January 22nd

FRQ Review Inference About Means

Due Thursday, January 23rd

Due Today: HWK 5.2 Review of 1-Sample Z & T-Tests & Confidence Intervals

Assigned: Monday, January 6th

Due Tomorrow: CWK 5.6 Paired T-test

Assigned: Today

Highly Recommended: You are able to answer Death Packet Problems 1-43

Name:__________________________ CWK 5.6 Paired T-test Period:______

Due: Thursday January 9th

Often times parents of students have had bad experiences with Statistics courses and pass on their

anxieties to their children. To test whether actually taking AP Statistics decreases students’ anxieties

about Statistics, an AP Statistics instructor gave a test to rate student anxiety at the beginning and

end of his course. Anxiety levels were measured on a scale of 0-10. Here are the data for 16 randomly

chosen students from a class of 180 students:

Pre-course anxiety level 7 6 9 5 6 7 5 7 6 4 3 2 1 3 4 2

Post-course anxiety level 4 3 7 3 4 5 4 6 5 3 2 2 1 3 4 3

Difference (Post – Pre)

1. Do the data indicate that anxiety levels about Statistics decreases after students take AP

Statistics? Test an appropriate hypothesis and state your conclusion. 2. Explain the P-value in context

3. Create and interpret a 90% confidence interval.

Most people are definitely dominant on one side of their body – either right or left. For some sports

being able to use both sides is an advantage, such as batting in baseball or softball. In order to

determine if there is a difference in strength between the dominant and non-dominant sides, a few

switch-hitting members of some school baseball and softball teams were asked to hit from both sides of

the plate during batting practice. The longest hit (in feet) from each side was recorded for each player.

The data are shown in the table at the right. Does this sample indicate that there is a difference in the

distance a ball is hit by batters who are switch-hitters?

4. Test an appropriate hypothesis and state your conclusion.

Dominant Non-

dominant Side Side

142 119 144 118 153 126 148 119 146 121 149 125 138 116 145 120 153 124 160 138 163 135 170 144 169 142 151 128 152 131 167 141 164 140 165 140 163 138

5. Create and interpret a 95% confidence interval.

6. Explain the meaning of the confidence level

Name:__________________________ Death Packet: Means & Errors Period:______

(Due Wednesday, January 22nd)

1. A guidance counselor is interested in comparing GPAs of students with home access to the internet

students who do not have this access. She pull the files of an SRS of ten students who do have home

access to the Internet and an SRS of ten who do not, and proceeds to run a t-test to compare the

mean GPAs of each group. Which of the following is a necessary assumption?

(A) The population standard deviations from each group are known.

(B) The population standard deviations from each group are unknown.

(C) The population standard deviations from each group are equal.

(D) The population of GPA scores from each group is normally distributed.

(E) The samples must be independent samples, and for each sample np and n (1 - p) must be at

least 10.

2. Which of the following is a true statement?

(A) Tests of significance (hypothesis tests) are designed to measure the strength of evidence

against the null hypothesis.

(B) A well-planned test of significance should result in a statement either that the null hypothesis is

true or that it is false.

(C) The null hypothesis is one-sided and expressed using either < or > if there is interest in

deviations in only one direction.

(D) When a true parameter value is farther from the hypothesized value, it becomes easier to

reject the alternative hypothesis.

(E) Increasing the sample size makes it more difficult to conclude that an observed difference

between observed and hypothesized values is significant.

3. Two confidence interval estimates from the same sample are (72.2, 77.8) and (71.3, 78.7). One

estimate is at the 95 percent level, and the other is at the 99 percent level. Which is which?

(A) (72.2, 77.8) is the 95 percent level.

(B) (72.2, 77.8) is the 99 percent level.

(C) This question cannot be answered without knowing the sample size.

(D) This question cannot be answered without knowing the sample standard deviation

(E) This question cannot be answered without knowing both the sample size and standard deviation.

4. Two 95 percent confidence interval estimates are obtained: 1st (78.5, 84.5) and 2nd (80.3, 88.2).

a. If the sample sizes are the same, which has the larger standard deviation?

b. If the sample standard deviations are the same, which has the larger sample size?

(A) a. 1st b. 1st

(B) a. 1st b. 2nd

(C) a. 2nd b. 1st

(D) a. 2nd b. 2nd

(E) More information is needed to answer these questions.

5. In general, how does doubling the sample size change the confidence interval size?

(A) Doubles the interval size

(B) Halves the interval size

(C) Multiplies the interval size by 1.414

(D) Divides the interval size by 1.414

(E) This question cannot be answered without knowing the sample size.

6. A confidence interval estimate is determined from the summer earnings of an SRS of n students. All

other things being equal, which of the following will result in a smaller margin of error?

(A) A greater confidence level

(B) A larger sample standard deviation

(C) A larger sample size

(D) Accepting less precision

(E) Introducing bias into sampling

7. Which of the following statements is true?

(A) It is helpful to examine your data before deciding to use a one-sided or a two-sided hypothesis

test.

(B) If the p-value is .05, the probability that the null hypothesis is correct is .05.

(C) The larger the p-value, the more evidence there is against the null hypothesis.

(D) If the p-value is small enough, we can conclude the alternative hypothesis is true.

(E) We do not use sample statistics in stating the hypothesis.

8. Which of the following is a true statement?

(A) The p-value of a test is the probability of obtaining a result as extreme as or more extreme than

the one obtained, assuming the null hypothesis is true.

(B) If the p-value for a test is .043, the probability that the null hypothesis is true is .043

(C) When the null hypothesis is rejected, it is because it is not true.

(D) The greater the p-value, the greater the evidence against the null hypothesis.

(E) A very large p-value provides convincing evidence that the null hypothesis is true.

9. Which of the following statement is true?

(A) If a population parameter is known, there is no reason to run a hypothesis test on that parameter.

(B) The P-value can be negative or positive depending upon whether the sample statistic is less than

or greater than the claimed value of the population parameter in the null hypothesis.

(C) The P-value is based on a specific test statistic, so it must be chosen before an experiment is

conducted.

(D) If a P-value is larger than a specified value ∝, then the data are statistically significant at that

level.

(E) The P-value is a probability calculated assuming that the alternative hypothesis is true.

10. The claim is made that the average IQ of student at Lake Wobegon High School is greater than 100.

To Test this, the principal gathers an SRS of 25 students and finds their mean IQ is 102 with a

standard deviation of 8.3 What is the test statistics in performing a hypothesis test of H0: 𝜇 = 100

and HA: 𝜇 > 100?

(A) T = 102−100

8.3

(B) T = 102−100

√8.3

25

(C) T = 102−100

√8.3

24

(D) T = 102−100

(8.3

√25)

(E) T = 102−100

(8.3

√24)

11. Which of the following would result in the widest confidence interval?

(A) Small sample size and 95 percent confidence

(B) Small sample size and 99 percent confidence

(C) Large sample size and 95 percent confidence

(D) Large sample size and 99 percent confidence

(E) This cannot be answered without knowing an appropriate standard deviation

12. Using the same data, one student performs a test H0: p = .85 and HA: p ≠ .85; a second student

performs a test H0: p = .85 and HA: p < .85. Even though both use 𝛼 = .05 level of significance, the

first student claims there is not enough evidence to reject H0. Which of the following could have

been the value of the test statistic?

(A) z = -2.3

(B) z = -1.8

(C) z = -1.3

(D) z = 1.3

(E) z =-1.8

13. A company employs ten executives each with a supporting secretary. The HR department wishes to

determine the difference in mean salaries between these executives and their secretaries. Based on

the most recent pay slips, a calculation is made for mean and standard deviation of the ten executive

salaries and similarly for the ten secretary salaries. A-t distribution is used to calculate a

confidence interval for the difference in the means. What is the principal mistake made in this

calculation?

(A) There is no reason why the population salaries can be assumed to be normal.

(B) There is no reason why the population salaries can be assumed to be independent.

(C) There is no indication of a lack of outliers in the respective populations.

(D) A one sample confidence interval using the set of differences is more appropriate.

(E) Using data from an entire population makes finding a confidence interval appropriate.

14. An electronic store chain executive claims that an average of 52.8 plasma television sets are being

sold per store per month. A stockholder believes this claim is high and randomly samples 30 stores.

What conclusion is reached if this sample mean is 49.9 with a standard deviation of 15.4?

(A) There is sufficient evidence to prove the executive’s claim is true.

(B) There is sufficient evidence to prove the executive’s claim is false.

(C) The stock holder has sufficient evidence to reject the executive’s claim.

(D) The stockholder does not have sufficient evidence to reject the executive’s claim.

(E) There is not sufficient data to reach any conclusion.

15. The prescription drugs Lipitor and Zetia are both known to lower blood cholesterol levels. In one

double-blind study Lipitor outperformed Zetia. The 95 percent confidence interval estimate of the

difference in mean cholesterol level lowering was (21, 37). Which of the following is a reasonable

conclusion?

(A) Zetia lowers cholesterol an average of 21 points, while Lipitor lowers cholesterol an average of 37

points.

(B) There is a .95 probability that Lipitor will outperform Zetia in lowering the cholesterol level of

any given individual.

(C) There is a .95 probability that Lipitor will outperform Zetia by at least 16 points in lowering the

cholesterol level of any given individual

(D) We should be 95 percent confident that Lipitor will outperform Zetia as a cholesterol-lowering

drug.

(E) None of the above.

16. A real estate agent claims that the mean sale price for homes in a city is $258,000. A building

contractor runs a check on a sample of ten recent sales and notes the average sale was $245,000

with a standard deviation of $38,000. What is the test statistic in performing a hypothesis of

H0: 𝜇 = 258,000 and HA: 𝜇 ≠ 258,000?

(A) t = 245,000−258,000

√38,000

10−1

(B) t = 2 (245,000−258,000

√38,00010

)

(C) t = 2(245,000−258,000

(38,000√10−1

))

(D) t = 245,000−258,000

38,000

√10

(E) t = 245,000−258,000

√38,000

10

17. A survey was conducted involving 50 of the 12,000 families living in a town. In 2006, the average

amount of medical expenses paid per family in the sample was $1,525 with a standard deviation of

$430. Establish a 95 percent confidence interval estimate for the total medical expenses paid by all

families in the town.

(A) $1,525 ± $122

(B) $76,250 ± $122

(C) $76,250 ± $6,110

(D) $18,300,000 ± $6,110

(E) $18,300,000 ± $1,470,000

18. In testing the null hypothesis H0: 𝜇 = 75 against the alternative hypothesis HA: 𝜇 > 75, a sample from

a normal population has a mean of 76.9 with a corresponding t-score of 2.23 and a p-value of .0175.

Which of the following is not a reasonable conclusion?

(A) If the null hypothesis is assumed to be true, the probability of obtaining a sample mean as

extreme as or more extreme than 76.9 is only .0175.

(B) In all samples using the sample size and the same sampling technique, the null hypothesis will be

wrong 1.75 percent of the time.

(C) At a significance level of 5 percent, there is sufficient evidence to reject the null hypothesis.

(D) If the population standard deviation 𝜎 had been known, then the z-score of 2.23 would have

resulted in a smaller p-value.

(E) If the alternative hypothesis had been two sided, then the same t-score would have resulted in a

p-value of 2 × .0175.

19. To determine the average spent on entertainment during a year in college, a simple random sample of

35 students is interviewed, showing a man of $825 with a standard deviation of $240. Which of the

following is the best interpretation of a 90 percent confidence interval estimate for the average

spent on entertainment during a year in college?

(A) 90 percent of college students spend between $756 and $894 on entertainment yearly.

(B) 90 percent of college students spend a mean dollar amount on entertainment yearly that is

between $756 and $894.

(C) We are 90 percent confident that college students spend between $756 and $894 on

entertainment yearly.

(D) We are 90 percent confident that college students spend a mean dollar amount between $756

and $894 on entertainment yearly.

(E) We are 90 percent confident that in the chosen sample, the mean dollar amount spent on

entertainment yearly by college students is between $756 and $894.

20. What is the critical t-value for finding a 96 percent confidence interval estimate from a sample of

18 observations?

(A) 2.054

(B) 2.205

(C) 2.214

(D) 2.224

(E) 2.235

21. The mean thrust of a certain model jet engine is 8,400 pounds. Concerned that a production process

change might have lowered the thrust, an inspector tests a random sample of five units, calculating a

man of 8,250 pounds with a t-score of -2.13 and a p-value of .0501. Which of the following is the

most reasonable conclusion?

(A) 94.99 percent of the engines produced under the new process will have a thrust under 8,250

pounds.

(B) 94.99 percent of the engines produced under the new process will have a thrust under 8,400

pounds.

(C) 5.01 percent of the engines produced under the new process will have a thrust over 8,400 pounds.

(D) There is evidence to conclude that the new process is producing engines with a mean thrust under

8,250 pounds.

(E) There is evidence to conclude that the new process is producing engines with a mean thrust under

8,400 pounds.

22. The weight of a “low-dosage” aspiring tablet is 81 mg according to the bottle label. An FDA inspector

weighs an SRS of eight tablets, obtains weights of 80, 81, 83, 81, 78, 81, and 80 mg, and runs a

hypothesis test of the manufacturer’s claim. Which of the following gives the p-values of this test?

(A) P(t < -1.271) with df = 7

(B) 2P(t < -1.271) with df = 7

(C) P(t < -1.271) with df = 8

(D) 2P(t < -1.271) with df = 8

(E) 0.5P(t < -1.271) with df = 8

23. City Planners are trying to decide various parking plan options ranging from more on street spaces to

multilevel parking garages. Before they make a decision, they wish to test the downtown merchant’s

claim that shoppers park for an average of only 38 minutes. The planners have decided to tabulate

parking durations for an SRS of 100 shoppers and reject the merchants’ claim if the sample mean

exceeds 40 minutes. If the merchants’ claim is wrong, and the true mean is 43 minutes, what is the

probability that the random sample will lead to a mistaken failure to reject the merchants’ claim?

Assume that the standard deviation in parking durations is 12 minutes.

(A) P(t <40−38

12

√100

)

(B) P(t >40−38

12

√100

)

(C) P(t <40−43

12

√100

)

(D) P(t >40−43

12

√100

)

(E) P(t >43−38

12

√100

)

24. A principal is informed that among the 1,250 students in her high school, the average GPA is 2.13

with a standard deviation of 0.61. With what margin of error is the mean GPA of these students

known?

(A) 0

(B) 0.61

(C) 0.61

√1,250

(D) 1.96(0.61

√1,250)

(E) None of the above gives the correct answer.

25. A researcher believes that a new diet should improve weight gain in laboratory mice. If the average

gain for 12 mice on the new diet is 5.3 ounces with a standard deviation of 0.5 ounces, while 15

control mice on the old diet gain an average of 5 ounces with a standard deviation of 0.4 ounces,

where is the p-value?

(A) Below .01

(B) Between .01 and .025

(C) Between .025 and .05

(D) Between .05 and .10

(E) Over .10

26. Suppose (48, 65) is a 95 percent confidence interval estimate for a population mean 𝜇. Which of the

following is a true statement?

(A) There is a .95 probability that 𝜇 is between 48 and 65.

(B) If 100 random samples of the given size are picked and a 95 percent confidence interval estimate

is calculated from each, then 𝜇 will be in 95 of the resulting intervals.

(C) If 95 percent confidence intervals are calculated from all possible samples of the given size, 𝜇 will be in 95 percent of these intervals.

(D) The probability that 𝜇 is in any particular interval can be any value between 0 and 1.

(E) Confidence level cannot be interpreted until after data is obtained.

27. Four math majors received the following salary offers upon graduation: $48,000, $55,000, $42,000

and $51,000. Assuming all assumptions are met, establish a 95 percent confidence interval for the

population mean.

(A) 49,000 ± 1.96 (5,477

√3)

(B) 49,000 ± 2.776 (5,477

√3)

(C) 49,000 ± 3.182 (5,477

√3)

(D) 49,000 ± 2.776 (5,477

√4)

(E) 49,000 ± 3.182 (5,477

√4)

28. A police department public relations spokesperson claims that the mean response time to a 911 call is

9 minutes. A newspaper reporter suspects that the response time is actually longer and runs a test

by examining the records of a random sample of 64 such calls. What conclusion is reached if the

sample mean is 9.55 minutes with a standard deviation of 3.00 minutes?

(A) The p-value is less than .001, indicating very strong evidence against the 9-minute claim.

(B) The p-value is .01, indicating strong evidence against the 9-minute claim.

(C) The p-value is .07, indicating some evidence against the 9-minute claim.

(D) The p-value is .18, indicating very little evidence against the 9-minute claim

(E) The p-value is .43, indicating no evidence against the 9-minute claim.

29. The number of 911 calls per day in a small Midwestern town is noted for a sample of 60 days with

�̅� = 23.4 and s = 3.7. With what degree of confidence can we assert that the mean number of 911

calls per day in this town is between 22.4 and 24.4?

(A) 48 percent

(B) 90 percent

(C) 95 percent

(D) 96 percent

(E) 99 percent

30. A quality inspector plans to test a sample of 40 bottles of soft drink labeled as 20 ounces. If she

finds the mean content to be less than 19.8 ounces, she will have the bottling machinery stopped and

recalibrated. If it is known that the machine operates with a standard deviation of 0.75 ounces,

what is the probability that the inspector will mistakenly stop a machine that is delivering a mean of

at least 20 ounces per bottle?

(A) 0.23

(B) .046

(C) .050

(D) .092

(E) .119

31. Under what conditions would it be meaningful to construct a confidence interval estimate when the

data consists of the entire population?

(A) If the population is small (n < 30)

(B) If the population is large (n ≥ 30)

(C) If a higher level of confidence is desired

(D) If the population is truly random

(E) Never

32. When making an inference about a population mean, which of the following suggests the use of

z-scores rather than t-scores?

(A) Outliers are absent

(B) The population is normal

(C) The sample size is under 30

(D) Strong skew is absent

(E) The Variance of population is known

33. A congressional representative serving on the Joint Committee on Taxation states that the average

yearly charitable contributions for taxpayers is $1,250. A lobbyist for a national church organization

who believes that the real figure is lower samples 12 families and comes up with a mean of $1,092

and a standard deviation of $308. Where is the p-value?

(A) Below .o1

(B) Between .01 and .025

(C) Between .025 and .05

(D) Between .05 and .10

(E) Over .10

34. In a one-sided hypothesis test for the mean, for a random sample of size 15 the t-scores of the

sample mean is 2.615. Is this significant at the 5 percent level? At the 1 percent level?

(A) Significant at the 1 percent level but not at the 5 percent level

(B) Significant at the 5 percent level but not at the 1 percent level

(C) Significant at both the 1 percent and 5 percent levels

(D) Significant at neither the 1 percent nor the 5 percent levels

(E) Cannot be determined from the given information.

35. A fitness center advertises that the average pulse rate of its members is 68.4bpm. A high school AP

statistics instructor suspects this is a made-up number and runs a hypothesis test on an SRS of 48

members, calculating a mean of 71.0 with a standard deviation of 10.3 bpm. In which of the following

intervals is the p-value located?

(A) P < .01

(B) .01 < P < .02

(C) .02 < P < .05

(D) .05 < P < .10

(E) P > .10

36. The 6.779 km annual boat race between Oxford and Cambridge is watched by an estimated quarter

million fans from the banks of the river. The average weight of these crew members has been

around 180 lb., but a U.S. sports statistician believes that the average weight of all collegiate crew

athletes in the United States is less than this. In a random sample of 50 U.S. crew members, a

hypothesis test with H0: 𝜇 = 180 and HA: 𝜇 < 180 results in a P-value of .078. Using this data, among

the following, which is the largest level of confidence for a two-sided confidence interval that does

not contain 180?

(A) 84 percent

(B) 90 percent

(C) 92 percent

(D) 94 percent

(E) 96 percent

37. A national parent organization is concerned that middle school students are spending too much time

(more than 8 hours a week) with home video game consoles like the Xbox 360, Wii, and PlayStation3.

A study is conducted sampling 150 students, and the resulting sample mean is 7.95 hours. What is

the appropriate alternative hypothesis for this study?

(A) HA: 𝜇 < 8

(B) HA: 𝜇 > 8

(C) HA: 𝜇 < 7.95

(D) HA: 𝜇 > 7.95

(E) HA: 𝜇 ≠ 7.95

38. Suppose fastball speeds in collegiate baseball are normally distributed with a mean of 86 mph and a

standard deviation of 4 mph. Suppose further that major league scouts will only consider pitching

prospects who can throw a fastball at least 90mph. what percentage of the pitching prospects who

are considered can throw at speeds over 95mph?

(A) 1.2 percent

(B) 7.7 percent

(C) 14.7 percent

(D) 15.9 percent

(E) 44.4 percent

39. In 1798 Henry Cavendish used a torsion balance to estimate the density of the earth at 5.42 times

the density of water. A modern day geologist runs a hypothesis test with Ha ≠ 5.42 and obtained a P-

value of .08. If the geologist had run a one-sided test with the same data, what is true about the

possible resulting p-value(s)?

(A) The only possible p-value is .04

(B) The only possible p-value is .08

(C) The only possible p-value is .16

(D) The possible p-values are .04 and .92

(E) The possible p-values are .04 and .96

40. Car insurance policies for teenagers have higher premiums than for adult drivers because teenagers

are considered to be a high-risk population. Suppose the average yearly cost of teenage (ages 16 to

19) insurance is $3,025 with a standard deviation of $430. Is there sufficient information to

answer either or both of the questions?

I. What is the probability that a randomly chosen teenage driver pays over $3,0000 a year for auto

insurance

II. What is the probability that the average amount paid in an SRS of 50 teenage drivers is more

than $3,000 a year?

(A) Insufficient information to answer either question.

(B) Sufficient information to answer question I, but not question II.

(C) Sufficient information to answer question II, but not question I.

(D) Sufficient information to answer both questions.

(E) With a sample of size 50, the sampling distribution is approximately normal with mean $3,025

and standard deviation of $430

√50

41. In a random sample of 50 mathematics majors at a large state university, their mean GPA on a 4

point scale was 3.65 with a standard deviation of 0.25. Which of the following is most likely true

about the distribution of GPAs among math majors at this university?

(A) The Central Limit Theorem guarantees that the distribution is approximately normal.

(B) The sample size of 50 guarantees that the distribution is approximately normal.

(C) Since the maximum possible GPA of 4.0 would have a z-score of 1.4 the distribution is skewed

left.

(D) Since the maximum possible GPA of 4.0 would have a z-score of 1.4, the distribution is skewed

right.

(E) Given that the subjects are math majors, the distribution is probably chi-square.

42. The 40 yard dash is an important diagnostic evaluation tool for all positions in both collegiate and

professional football recruiting. Which of the following is the best interpretation of a 95 percent

confidence interval of (4.67, 4.87) seconds for the mean 40 yard dash time?

(A) Ninety-five percent of all players run the 40 yard dash in between 4.67 and 4.87 seconds.

(B) Ninety-five percent of all players have a mean 40 yard dash time between 4.67 and 4.87 seconds.

(C) Players run the 40 yard dash in between 4.67 and 4.87 seconds 95 percent of the time.

(D) We are 95 percent confident that for any given player his 40 yard dash time is between 4.67

and 4.87 seconds.

(E) We are 95 percent confident that the mean 40 yard dash time is between 4.67 and 4.87

seconds.

43. Research has shown that tornado wind speeds are normally distributed with a mean of 112 mph.

Consider random samples of 10 tornados each.

The variable �̅�−112

𝑠

√10

(A) Has a normal distribution

(B) Has a t-distribution with df = 9

(C) Has a t-distribution with df = 10

(D) Has a chi-square distribution

(E) Has a distribution that cannot be specified because the sample size is too small (n < 30)

44. An assembly line machine is supposed to turn out bowling balls with a diameter of 8.55 inches. Each

day an SRS of five balls are pulled and measured. If there mean diameter is under 8.35 inches or

over 8.75 inches, the machinery is stopped and an engineer is called to make adjustments before

production is resumed. The quality control procedure may be viewed as a hypothesis test with

H0: 𝜇 = 8.55 and HA: 𝜇 ≠ 8.55. What would a Type II error result in?

(A) A warranted halt in production to adjust the machinery

(B) An unnecessary stoppage of the production process

(C) Continued production of wrong size bowling balls

(D) Continued production of proper size bowling balls

(E) Continued production of bowling balls that randomly are the right or wrong size

45. When leaving for school on an overcast morning, you make a judgment on the null hypothesis: The

weather will remain dry. What would the results be of Type I and Type II errors?

(A) Type I error: get drenched. Type II error: needlessly carry around an umbrella.

(B) Type I error: needlessly carry around an umbrella. Type II error: get drenched

(C) Type I error: carry an umbrella, and it rains. Type II error: carry no umbrella, but weather

remains dry.

(D) Type I error: get drenched. Type II error: carry no umbrella, but weather remains dry.

(E) Type I error: get drenched. Type II error: carry an umbrella, and it rains.

46. What is the probability of a Type II error when a hypothesis test is being conducted at the 5

percent significance level (α= .05)?

(A) .05

(B) .10

(C) .90

(D) .95

(E) There is insufficient information to answer this question.

47. Which of the following is a true statement?

(A) The probability of a Type II error does not depend on the probability of a Type I error.

(B) In conducting a hypothesis test, it is possible to simultaneously make both a Type I and Type II

error.

(C) A Type II error will result if one incorrectly assumes the data are normally distributed

(D) Type I and II errors are caused by mistakes, however small, by the person conducting the test.

(E) A Type I error is a conditional probability.

48. A manufacturer of heart-lung machines periodically checks a sample of its product and performs a

major recalibration if readings are sufficiently off target. Similarly, a rug factory periodically

checks the sizes of its throw rugs coming off an assembly line and halts production if measurements

are sufficiently off target. In both situations, we have the null hypothesis that the production

equipment is performing satisfactorily. For each situation, which is the more serious concern, a Type

I or Type II error?

(A) Machine producer: Type I error, carpet manufacturer: Type I error

(B) Machine producer: Type I error, carpet manufacturer: Type II error

(C) Machine producer: Type II error, carpet manufacturer: Type I error

(D) Machine producer: Type II error, carpet manufacturer: Type II error

(E) This is impossible to answer without making an expected value judgment between human life and

accurate throw rug sizes.

49. Given and experiment H0: 𝜇 = 25 and HA: 𝜇 > 25, and a possible true population value of 26, which of

the following increase with an increase in the sample size n?

(A) The probability of a Type I error

(B) The probability of a Type II error

(C) The power of the test

(D) The significance level 𝛼

(E) 1 – power

50. Which of the following statements is incorrect?

(A) The significance level of a test is the probability of a Type II error.

(B) Given a particular alternative, the power of a test against that alternative is 1 minus the

probability of the Type II error associated with that alternative.

(C) If the significance level remains fixed, increasing the sample size will reduce the probability of a

Type II error.

(D) If the significance level remains fixed, increasing the sample size will raise the power.

(E) Holding the sample size fixed, increasing the significance level will decrease the probability of a

Type II error.

51. Which of following is incorrect?

(A) The power of a test concerns its ability to detect a true alternative hypothesis.

(B) The significance level of a test is the probability rejecting a true null hypothesis.

(C) The probability of a Type I error plus the probability of A Type II error always equals 1.

(D) Power equals 1 minus the probability of failing to reject a false null hypothesis.

(E) Anything that makes a null hypothesis harder to reject will increase the probability of committing

a Type II error.

52. Suppose H0: p = .6, HA: p > .6, and against the alternative p = .7, the power is .8. Which of the

following is a valid conclusion?

(A) The probability of committing a Type I error is .1.

(B) If p = .7 is true, the probability of failing to reject H0 is .2.

(C) The probability of committing a Type II error is .3.

(D) All of the above are valid conclusions.

(E) None of the above are valid conclusions.

53. A company will market a new hybrid luxury car only if they can sell it for more than $50,000

(otherwise, it will lose money). They do a random survey of 50 potential customers and run a

hypothesis test with H0: 𝜇 = 50,000 and HA: 𝜇 > 50,000. What would be the consequences of Type I

and Type II errors?

(A) Type I error: produce a non-profitable car; Type II error: fail to produce a profitable car

(B) Type I error: fail to produce a profitable car; Type II error: produce a non- profitable car

(C) Type I error: fail to produce a non-profitable car; Type II error: produce a profitable car

(D) Type I error: fail to produce a profitable car; Type II error: produce a profitable car

(E) Type I error: produce a non-profitable car; Type II error: fail to produce a non-profitable car

54. A company that produces paper towels continually monitors wet towel strength. If the mean

strength from a sample drops below a specified level, the production process is halted, and the

machinery inspected. Which of the following would result from a Type I error?

(A) Halting the production process when sufficient customer complaints are received.

(B) Halting the production process when the wet towel strength is not within specifications

(C) Halting the production process when the wet towel strength is within specifications

(D) Allowing the production process to continue when the wet towel strength is below specifications

(E) Allowing the production process to continue when the wet towel strength is within specifications

55. Which of following is a true statement?

(A) A p-value is a conditional probability.

(B) The p-value is the probability that the null hypothesis is true.

(C) A p-value is the probability the null hypothesis is true given a particular observed statistic.

(D) A p-value is the probability of Power.

(E) Large p-values are evidence against the null hypothesis because they say that the observed result

is unlikely to occur when the null hypothesis is true.

56. If all other variables remain constant, which of the following will not increase the power of a

hypothesis test?

(A) Increasing the sample size

(B) Increasing the significance level

(C) Increasing the probability of a Type II error

(D) Decreased variability in the data

(E) Increased distance between the true and the hypothesized parameter.

57. Given that the power of a significance test against a particular alternative is 96 percent, which of

the following is true?

(A) The probability of mistakenly rejecting a true null hypothesis is less than 4 percent.

(B) The probability of mistakenly rejecting a true null hypothesis is 4 percent.

(C) The probability of mistakenly rejecting a true null hypothesis is greater than 4 percent.

(D) The probability of mistakenly failing to reject a false null hypothesis is 4 percent.

(E) The probability of mistakenly failing to reject a false null hypothesis is different from 4 percent.

58. A government statistician claims that the mean income level of families living in subsidized housing is

$9,250 with a standard deviation of $2,575. A reporter plans to test this claim through interviews

with a random sample of 50 families. If she finds a sample mean more than $500 different from the

claimed$9,250, she will dispute the statisticians claim. What is the probability that the reporter will

mistakenly reject a true claim?

(A) .043

(B) .085

(C) .170

(D) .830

(E) .915

59. It is hypothesized that players lose an average of $110 during an evening at a blackjack table in a

Las Vegas casino. A government tax collector believes the true mean loss is greater than this claim

and plans a hypothesis test on a random sample of blackjack players. Suppose the tax collector is

correct. For a sample of n = 100 and a significance level of 𝛼 = .05, for which of the following

possible actual mean losses will the power of the test be the greatest?

(A) 95

(B) 100

(C) 105

(D) 115

(E) 120

60. Consider a hypothesis test with H0: 𝜇 = 58 and HA: 𝜇 > 58. Which of the following choices of

significance level and sample size results in the greatest power of the test when 𝜇 = 60

(A) 𝛼 = 0.05, n = 20

(B) 𝛼 = 0.01, n = 20

(C) 𝛼 = 0.05, n = 25

(D) 𝛼 = 0.01, n = 25

(E) There is no way of answering without knowing the strength of the given power.

61. Last year it was calculated that recreation expenditures were 8.7 percent of total personal

consumption. An economist believes that this percentage is going up as we move further out of the

recession. Consider a hypothesis test with H0: p = .087 and HA: p > .087. Which of the following

choices of significance level and sample size results in the greatest power of the test when p =.095?

(A) 𝛼 = .05, n = 20

(B) 𝛼 = .10, n = 20

(C) 𝛼 = .05, n = 30

(D) 𝛼 = .10, n = 30

(E) Unknown without knowledge of the strength of the given power.

62. In a comparison of the life expectancies of two models of washing machines, the average years

before breakdown in an SRS of 10 machines of one model, which is compared with that of 15

machines of a second model. The 95 percent confidence interval estimate of the difference is

(6, 12). Which of the following is the most reasonable conclusion?

(A) The mean life expectancy of one model is twice that of the other

(B) The mean life expectancy of one model is 6 years, while the mean life expectancy of the other is

12 years.

(C) The probability the life expectancies are different is .95.

(D) The probability the difference in life expectancies is greater than 6 years is .95.

(E) We should be 95 percent confident that the difference in life expectancies is between 6 and 12

years.

63. Men and women were surveyed as to the ideal number of children a family should have. For 500 men,

the mean was 2.95 with a standard deviation of 1.65; for 450 women, the mean was 3.05 with a

standard deviation of 1.80. Assuming all assumptions are met, establish a 90 percent confidence

interval for the difference in population means.

(A) -0.10 ± 1.96√(2.95)2

500+

(3.05)2

450

(B) -0.10 ± 1.645(1.65

√500+

1.80

√450)

(C) -0.10 ± 1.645√(1.65)2

500+

(1.80)2

450

(D) -0.10 ± 1.96(2.95

√500+

1.80

√450)

(E) -0.10 ± 3.29(1.65

√500+

1.80

√450)

64. First-year high school students were randomly put into either traditional math classes or math

classes taught with a new innovative approach. In a standard algebra test, 230 traditionally taught

students scored an average of 76.5 with a standard deviation of 7.4, while 185 students under the

new approach averaged 78.8 with a standard deviation of 5.6. Is there sufficient evidence of a

difference in the test results-what is the test statistic for H0: 𝜇1 − 𝜇2 = 0 and HA: 𝜇1 − 𝜇2 ≠ 0?

(A) t = 76.5−78.8

√(7.4)2

230+

(5.6)2

185

(B) t = 2 76.5−78.8

√(7.4)2

230+

(5.6)2

185

(C) t = 76.5−78.8

(7.4

√230+

5.6

√185)

(D) t = 2 76.5−78.8

(7.4

√230+

5.6

√185)

(E) t = 76.5−78.8

(6.5

√207.5)

65. On a windless day, a member of a high school’s football team, its punting specialists, kicks tow

identical footballs, one filled with helium and one filled with air, ten times each. He alternates

footballs and is not told there is any difference. The distance (in yards) of the kicks are recorded

in the table below.

Helium 36 40 37 34 35 35 42 36 31 36

Air 38 37 31 36 40 34 37 36 34 34

Assuming that all assumptions for inference are met, which test should be used to determine if

there is a statistical difference in yards kicked using the two balls?

(A) A one-sample paired z-test on a set of differences

(B) A one sample paired t-test on a set of differences

(C) A two-sample z-test

(D) A two sample t-test

(E) A linear regression t-test with df = 10 - 2 = 8

66. To test whether husbands or wives have greater manual agility, an SRS of 50 married couples is

chosen, and all 100 people are given a 1-minute period to find and place strangely shaped pegs into

matching holes. What is the conclusion at a 5 percent significance level if a two-sample hypothesis

test, H0: 𝜇1 − 𝜇2 = 0 and HA: 𝜇1 − 𝜇2 ≠ 0 results in a p-value of 0.15.

(A) The observed difference between husbands and wives is significant.

(B) The observed difference is not significant.

(C) A conclusion is not possible without knowing the mean number of pegs placed by husbands and by

wives.

(D) A conclusion is not possible without knowing both the mean and standard deviation of the number

of pegs placed by husbands and by wives.

(E) A two-sample hypothesis test should not be used in this example.

67. Administrators working for a national sports association randomly picked 20 Division 1 and 30

Division 3 football programs and ran a two-sample inference to compare the mean GPA of football

players in the two divisions. With H0: 𝜇𝐷1 = 𝜇𝐷3 and HA: 𝜇𝐷1 < 𝜇𝐷3, a P-value less than .05

resulted. Which of the following is a proper conclusion?

(A) It cannot be concluded that the observed difference in mean GPAs is significant because the

same number of programs from each division was not included.

(B) It cannot be concluded that the observed difference in mean GPAs is significant because this

was an observational study, not an experiment.

(C) It cannot be concluded that the observed difference in mean GPAs is significant unless students

from the chosen programs were randomly selected.

(D) It cannot be concluded that the observed difference in mean GPAs is significant because

Division 2 programs were excluded from the study.

(E) It can be concluded that the observed difference in mean GPAs is significant.

68. A high school has six math teachers and six science teachers. When comparing their mean years of

service, which of the following is most appropriate?

(A) A two-sample z-test of population means

(B) A two-sample t-test of population means

(C) A one-sample z-test on a set of differences

(D) A one sample t-test on a set of differences

(E) None of the above are appropriate

69. Two pain relief medicines are tested on random samples of post-operative patients at the 5 percent

significance level as to whether or not mean durations of relief are different. The data give:

Brand Sample Size Mean Relief (min) Standard Deviation (min)

A 20 131 12

B 15 140 15

What is the conclusion of a two-sample t-test?

(A) P < .05 so reject H0

(B) P < .05 so fail to reject H0

(C) P > .05 so reject H0

(D) P > .05 so fail to reject H0

(E) The 5 percent significance level is inappropriate for medical decisions.

70. A college recruiter is interested in comparing the SAT math and verbal scores of applicants to the

college. An SRS of 40 applicants is chosen, and the math and verbal scores are noted. Which of the

following is a proper test?

(A) Test of difference in two population means.

(B) Test of difference in two population proportions.

(C) One sample test on difference of paired data.

(D) Chi-square goodness-of-fit test.

(E) Chi-square test of homogeneity.

71. Ten students were randomly selected from a high school to take part in a program designed to raise

their reading comprehension. Each student took a test before and after completing the program.

The mean of the differences between the score after the program and the score before the

program is 16. It was decided that all students in the school would take part in this program during

the next school year. Let A denote the mean score after the program and B denote the mean

score before the program for all students in the school. The 95 percent confidence interval

estimate of the true mean difference for all students is (9, 23). Which of the following statements

is a correct interpretation of this confidence interval?

(A) A > B with probability 0.95.

(B) A < B with probability 0.95.

(C) A is around 23 and B is around 9.

(D) For any A and B with (A − B) 14, the sample result is quite likely.

(E) For any A and B with 9 <(A − B) < 23, the sample result is quite likely.

72. The distribution of the weights of loaves of bread from a certain bakery follows approximately a

normal distribution. Based on a very large sample, it was found that 10 percent of the loaves

weighed less than 15.34 ounces, and 20 percent of the loaves weighed more than 16.31 ounces.

What are the mean and standard deviation of the distribution of the weights of the loaves of

bread?

(A) = 15.82, = 0.48

(B) = 15.82, = 0.69

(C) = 15.87, = 0.50

(D) = 15.93, = 0.46

(E) = 16.00, = 0.50

73. The manager of a factory wants to compare the mean number of units assembled per employee in a

week for two new assembly techniques. Two hundred employees from the factory are randomly

selected and each is randomly assigned to one of the two techniques. After teaching 100 employees

one technique and 100 employees the other technique, the manager records the number of units

each of the employees assembles in one week. Which of the following would be the most

appropriate inferential statistical test in this situation?

(A) One-sample z-test

(B) Two-sample t-test

(C) Paired t-test

(D) Chi-square goodness-of-fit test

(E) One-sample t-test

74. As lab partners, Sally and Betty collected data for a significance test. Both calculated the same

z-test statistic, but Sally found the results were significant at the = 0.05 level while Betty found

that the results were not. When checking their results, the women found that the only difference

in their work was that Sally used a two-sided test, while Betty used a one-sided test. Which of the

following could have been their test statistic?

(A) 1.980 (B) 1.690 (C) 1.340 (D) 1.690 (E) 1.780

FRQ Review Inference About Means

Due Thursday, January 23rd 2000 Number 2

1. Anthropologists have discovered a prehistoric cave dwelling that contains a large number of adult

human footprints. To study the size of the adults who used the cave dwelling, they randomly

selected 20 of the footprints from the population of all footprints in the cave and measured the

length of those footprints. Some statistics resulting from this random sample are as follows.

Sample size 20 Minimum 15.2 cm

Mean 24.8 cm First quartile 18.7 cm

Standard deviation 7.5 cm Median 21.5 cm

Third quartile 30.0 cm

Maximum 37.0 cm

The anthropologists would like to construct a 95 percent confidence interval for the mean foot

length of the adults who used the cave dwelling.

(a) What assumptions are necessary in order for this confidence interval to be appropriate?

(b) Discuss whether each of the assumptions listed in your response to (a) appears to be satisfied in

this situation.

2002 Number 5

2. Sleep researchers know that some people are early birds (E), preferring to go to bed by 10 p.m. and

arise by 7 a.m., while others are night owls (N), preferring to go to bed after 11 p.m. and arise after

8 a.m. A study was done to compare dream recall for early birds and night owls. One hundred people

of each of the two types were selected at random and asked to record their dreams for one week.

Some of the results are presented below.

a) The researchers believe the night owls may have better dream recall than do early birds. One

parameter of interest to the researchers is the mean number of dreams recalled per week with

μE representing this mean for early birds and μN representing this mean for night owls. The

appropriate hypothesis would then be Ho: μE – μN = 0 and Ha : μE – μN < 0. State two other pairs of

hypotheses that might be used to test the researchers’ belief. Be sure to define the parameter

of interest in each case.

b) Use the data provided to carry out a test of the hypothesis about the mean number of dreams

recalled per week given in the statement of part a). Do the data support the researchers’ belief?

2004 Number 6

3. A pharmaceutical company has developed a new drug to reduce cholesterol. A regulatory agency will

recommend the new drug for use if there is convinced evidence that the mean reduction in

cholesterol level after one month of use is more than 20 milligrams/deciliter (mg/dl), because a mean

reduction of this magnitude would be greater than the mean reduction for the current most widely

used drug.

The pharmaceutical company collected data by giving the new drug to a random sample of 50 people

from the population of people with high cholesterol. The reduction in cholesterol level after one

month of use was recorded for each individual in the sample, resulting in a sample mean reduction and

standard deviation of 24 mg/dl and 15 mg/dl, respectively.

(a) The regulatory agency decides to use an interval estimate for the population mean reduction in

cholesterol level for the new drug. Provide this 95 percent confidence interval. Be sure to

interpret this interval.

(b) Because the 95 percent confidence interval includes 20, the regulatory agency is not convinced

that the new drug is better than the current best-seller. The pharmaceutical company tested

the following hypothesis.

Ho: 𝜇 = 20 versus Ha: 𝜇 > 20,

where 𝜇 represents the population mean reduction in cholesterol level for the new drug.

The test procedure resulted in a t-value of 1.89 and a p-value of 0.033. Because the p-value was

less than 0.05, the company believes that there is convincing evidence that the mean reduction in

cholesterol level for the new drug is more than 20. Explain why the confidence interval and the

hypothesis test led to different conclusions.

(c) The company would like to determine a value L that would allow them to make the following

statement.

We are 95 percent confident that the true mean reduction in cholesterol level is greater

than L.

A statement of this form is call a one-sided confidence interval. The value of L can be found

using the following formula.

*s

L x tn

This has the same form as the lower endpoint of the confidence interval in part (a), but requires

a different critical value, t*. What value should be used for t*?

Recall that the sample mean reduction in cholesterol level and standard deviation are 24 mg/dl

and 15 mg/dl, respectively. Compute the value of L.

(d) If the regulatory agency had used the one-sided confidence interval in part (c) rather than the

interval constructed in part (a), would it have reached a difference conclusion? Explain.

Chapter 24

2006 Number 4

4. Patients with heart-attack symptoms arrive at an emergency room either by ambulance or self-

transportation provided by themselves, family, or friends. When a patient arrives at the emergency

room, the time of arrival is recorded. The time when the patient’s diagnostic treatment begins is

also recorded.

An administrator of a large hospital wanted to determine whether the mean wait time (time between

arrival and diagnostic treatment) for patients with heart-attack symptoms differs according to the

mode of transportation. A random sample of 150 patients with heart-attack symptoms who had

reported to the emergency room was selected. For each patient, the mode of transportation and

wait time were recorded. Summary statistics for each mode of transportation are shown in the table

below.

Mode of

Transportation

Sample

Size

Mean Wait Time

(in minutes)

Standard Deviation

of Wait Times

(in minutes)

Ambulance 77 6.04 4.30

Self 73 8.30 5.16

(a) Use a 99 percent confidence interval to estimate the difference between the mean wait times

for ambulance-transported patients and self-transported patients at this emergency room.

(b) Based only on this confidence interval, do you think the difference in the mean wait times is

statistically significant? Justify your answer.

Chapter 25

2004 Form B Question 4

5. The principal at Crest Middle School, which enrolls only sixth-grade students and seventh-grade

students, is interested in determining how much time students at that school spend on homework

each night. The table below shows the mean and standard deviation of the amount of time spent on

homework each night (in minutes) for a random sample of 20 sixth-grade students and a separate

random sample of 20 seventh-grade students at this school.

Mean Standard Deviation

Sixth-grade students 27.3 10.8

Seventh-grade students 47.0 12.4

Based on dotplots of these data, it is not unreasonable to assume that the distribution of times for

each grade were approximately normally distributed.

(a) Estimate the difference in mean times spent on homework for all sixth- and seventh-grade

students in this school using an interval. Be sure to interpret your interval.

(b) An assistant principal reasoned that a much narrower confidence interval could be obtained if the

students were paired based on the responses; for example, pairing the sixth-grade student and

the seventh-grade student with the highest number of minutes spent on homework, the sixth-

grade student and seventh-grade student with the next highest number of minutes spent on

homework, and so on. Is the assistant principal correct in thinking that matching students in this

way and then computing a matched-pairs confidence interval for the mean difference in time

spent on homework is a better procedure than the one used in part (a)? Explain why or why not.

2001 Number 5

6. A growing number of employers are trying to hold down the costs that they pay for medical insurance

for their employees. As part of this effort, many medical insurance companies are now requiring

clients to use generic brand medicines when filling prescriptions. An independent consumer advocacy

group wanted to determine if there was a difference, in milligrams, in the amount of active

ingredient between a certain “name” brand drug and its generic counterpart. Pharmacies may store

drugs under different conditions. Therefore, the consumer group randomly selected ten different

pharmacies in a large city and filled two prescriptions at each of these pharmacies, one for the

“name” brand and the other for the generic brand of the drug. The consumer group’s laboratory then

tested a randomly selected pill from each prescription to determine the amount of active ingredient

in the pill. The results are given in the following table.

ACTIVE INGREDIENT

(in milligrams)

Pharmacy 1 2 3 4 5 6 7 8 9 10

Name brand 245 244 240 250 243 246 246 246 247 250

Generic

brand

246 240 235 237 243 239 241 238 238 234

Based on these results, what should the consumer group’s laboratory report about the difference in

the active ingredient in the two brands of pills? Give appropriate statistical evidence to support your

response.

2005 Form B Number 6

7. Regulations require that product labels on containers of food that are available for sale to the public

accurately state the amount of food in those containers. Specifically, if milk containers are labeled

to have 128 fluid ounces and the mean number of fluid ounces of milk in the containers is at least

128, the milk processor is considered to be in compliance with the regulations. The filling machines

can be set to the labeled amount. Variability in the filling process causes the actual contents of milk

containers to be normally distributed. A random sample of 12 containers of milk was drawn from the

milk processing line in a plant, and the amount of milk in each container was recorded.

(a) The sample mean and standard deviation of this sample of 12 containers of milk were 127.2

ounces and 2.1 ounces, respectively. Is there sufficient evidence to conclude that the packaging

plant is not in compliance with the regulations? Provide statistical justification for your answer.

Inspectors decide to study a particular filling machine within this plant further. For this machine,

the amount of milk in the containers has a mean of 128.0 fluid ounces and a standard deviation of 2.0

fluid ounces.

(b) What is the probability that a randomly selected container filled by this machine contains at

least 125 fluid ounces?

(c) An inspector will randomly select 12 containers filled by this machine and record the amount of

milk in each. What is the probability that the minimum (smallest amount of milk) recorded in the

12 containers will be at least 125 fluid ounces? (Note: In order for the minimum to be at least

125 fluid ounces, each of the 12 containers must contain at least 125 fluid ounces.)

An analyst wants to use simulation to investigate the sampling distribution of the minimum. This

analyst randomly generates 150 samples, each consisting of 12 observations, from a normal

distribution with mean 128 and standard deviation 2 and finds the minimum for each sample. The 150

minimums (sorted from smallest to largest) are shown on the next page.

(d) Use the simulation results to estimate the probability that was requested in part (c) and compare

this estimate with the theoretical value you calculated.