# Mathematics

30. What is the probability of the subject obtaining his/her first correct guess within the first 4 questions?

A) 0.3164 C) 0.6836 B) 0.8945 D) 0.5781

Extra Credit 1. What is the probability it will take exactly 6 rolls of two fair dice to make a 7?

A) 0.9330 C) 0.0804 B) 0.0670 D) 0.9196

2. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals

8?

A) (0.6) 8 (0.4)

2

B)

10!

8! (0.6)

8 (0.4)

2

C) 45(0.6) 8 (0.4)

2

D) 45(0.6) 2 (0.4)

8

E) None of the above.

3. A college basketball player makes 6 5

of her free throws. Assume free throws are independent. What is the probability that she makes exactly three of her next four free throws?

A) ( ) ( )1

6

53

6 14

C) ( ) ( )3

6

51

6 14

B) ( ) ( )1

6

53

6 1

D) ( ) ( )3

6

51

6 1

4. A batch of 100 chocolate chip cookies contains 10 burnt cookies. Five cookies are chosen at random,

without replacement. Find the probability that the sample contains at least one burnt cookie. A) 0.3349 C) 0.4162 B) 0.6606 D) 0.5839

5. The exponential distribution is ______. A) symmetric B) bell-shaped C) All of the above D) None of the above

6. Although cities encourage carpooling to reduce traffic congestion, most vehicles carry only one

person. For example, nationally 75.5% of the people drive to work alone. a) If you choose 12 vehicles driving to work at random, what is the probability that more than half

(that is, 7 or more) carry just one person? b) If you choose 80 vehicles at random, what is the probability that more than half (that is, 41 or

more) carry just one person?

Chs 6 – 8

1. The lifetime (in hours) of a 60-watt light bulb is a random variable that has a Normal distribution with

σ = 30. A random sample of 25 bulbs put on test produced a sample mean lifetime of x = 1038. If in a study of the lifetime of 60-watt light bulbs it was desired to have a margin of error no larger than 6 hours with 99% confidence, how many randomly selected 60-watt light bulbs should be tested to achieve this result? A) 13 B) 97 C) 165 D) 42 E) Not within ±2 of any of the above.

Use the following to answer questions 2 and 3: A manufacturer of a specific part used in the operation of a gas turbine engine is concerned because the part is designated as critical-to-quality (CQT) and is costly to produce. The process that is used to produce the part has been studied extensively and has been shown to be stable and predictable for some lengthy period of time. When the process is in the stable state, the crucial measurement on the CQT part is known to be Normally distributed with a mean of µ = 1.58 cm and with a standard deviation of σ = 0.10 cm. In order to check on the status of the production process, a monitoring plan has been established, which requires that a sample of four manufactured parts should be selected at random each

hour; if the mean x of the sample exceeds 1.71 cm then the process is stopped, examined for possible problems, and repairs made to the process if needed.

2. If the process is operating as it should, what is the probability that this rule regarding x results in a shutdown when in fact there is nothing wrong with the process (this is called a false alarm)? A) 0.0013 B) 0.0094 C) 0.0986 D) 0.0047 E) Not within ± 0.001 of any of the above.

3. If, in fact, the process has changed and the process mean has shifted to µ = 1.73, what is the

probability that this rule regarding x will fail to detect the shift? A) 0.3446 B) 0.6554 C) 0.9987 D) 0.0013 E) Not within ± 0.001 of any of the above.

4. In the last mayoral election in a large city, 47% of the adults over the age of 65 voted Republican. A

researcher wishes to determine if the proportion of adults over the age of 65 in the city who plan to vote Republican in the next mayoral election has changed. Let p represent the proportion of the population of all adults over the age of 65 in the city who plan to vote Republican in the next mayoral election. In terms of p, the researcher should test which of the following null and alternative hypotheses? A) Ho: p = 0.47 vs. Ha: p < 0.47 B) Ho: p = 0.47 vs. Ha: p ≠ 0.47 C) Ho: p = 0.47 vs. Ha: p > 0.47

Use the following to answer questions 5 and 6: The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude, and study habits of college students. Scores range from 0 to 200 and follow (approximately) a Normal distribution with mean 115 and standard deviation 25. You suspect that incoming freshmen at your school have a mean µ  which is different from 115 because they are often excited yet anxious about entering college. To test your suspicion, you decide to test the hypotheses Ho: µ = 115 versus Ha: µ ≠ 115. You give the SSHA to 25 incoming freshmen and find their mean score to be 116.2. 5. What is the value of the test statistic?

A) z = 0.048 C) z = 1.2 B) z = 0.24 D) z = 1.96

6. What is the value of the p-value?

A) 0.1151 C) 0.4052 B) 0.2302 D) 0.8104

Use the following to answer questions 7 and 8: The attention span of little kids (ages 3–5) is claimed to be Normally distributed with a mean of 15 minutes and a standard deviation of 4 minutes. A test is to be performed to decide if the average attention span of these kids is really this short or if it is longer. You decide to test the hypotheses Ho: µ = 15 versus Ha: µ > 15 at the 5% significance level. A sample of 10 children will watch a TV show they have never seen before, and the time until they walk away from the show will be recorded. 7. Fill in the blank. At a significance level of 5%, the decision rule would be to reject the null hypothesis if

the observed sample mean is greater than _________ minutes. A) 15.66 C) 17.48 B) 17.08 D) 19

8. If, in fact, the true mean attention span of these kids is 18 minutes, what is the probability of a Type II

error? A) 0.0107 C) 0.3405 B) 0.2335 D) 0.7665

9. A study has been completed involving a test of significance of the null hypothesis Ho: µ = 0. The

researchers have discovered that the power of the test is too small. What can the researchers try to do in order to increase the power of their test procedure? A) Increase the level of significance, α . B) Increase the sample size, n. C) Decrease the population standard deviation, σ . D) All of the above. E) None of the above; nothing can be done to increase the power.

Use the following to answer questions 10 and 11: The time needed for college students to complete a certain paper-and-pencil maze follows a Normal distribution with a mean of 30 seconds and a standard deviation of 3 seconds. You wish to see if the mean time µ is changed by vigorous exercise, so you have a group of nine college students exercise vigorously for 30 minutes and then complete the maze. Assume that σ remains unchanged at 3 seconds. The hypotheses you decide to test are Ho: µ = 30 versus Ha: µ ≠ 30.

10. Suppose it takes the nine students an average of x = 32.05 seconds to complete the maze. At the 1% significance level, what can you conclude? A) Ho should be rejected because the p-value is less than 0.01. B) Ho should not be rejected because the p-value is greater than 0.01. C) Ha should be rejected because the p-value is less than 0.01. D) Ha should not be rejected because the p-value is greater than 0.01.

11. Suppose you compute the average time x that it takes these students to complete the maze and you find that the results are significant at the 5% level. What can you conclude? A) The test would also be significant at the 10% level. B) The test would also be significant at the 1% level. C) Both of the above. D) None of the above.

Use the following to answer questions 12–13: A 95% confidence interval (using the conservative value for the degrees of freedom) for

1 2 µ µ− , based on two independent samples of sizes 18 and 20, respectively, gives us (45.6, 56.7).

12. What was the observed difference between the two sample means 1x and 2x ? A) 11.1 C) 51.15 B) 45.6 D) 56.7

13. What would be the margin of error for a 99% confidence interval for 1 2

µ µ− ? A) 2.63 C) 5.55 B) 2.898 D) 7.62

14. When the sample size is very large, the corresponding t distribution is very close to the normal

distribution. A) True B) False

Use the following to answer questions 15–18: You wish to compare the prices of apartments in two neighboring towns. You take a simple random sample of 12 apartments in town A and calculate the average price of these apartments. You repeat this

for 15 apartments in town B. Let 1

µ represent the true average price of apartments in town A and 2

µ the average price in town B. 15. What would be the hypotheses for this problem?

A) Ho: 1

µ = 2

µ versus Ha: 1

µ < 2

µ

B) Ho: 1

µ = 2

µ versus Ha: 1

µ > 2

µ

C) Ho: 1

µ = 2

µ versus Ha: 1

µ ≠ 2

µ 16. If we were to use the pooled t test, what would be the degrees of freedom?

A) 11 C) 14 B) 12 D) 25

17. If we were to use the unpooled t test, what would be the conservative estimate for the degrees of

freedom? A) 11 C) 14 B) 12 D) 25

18. Suppose we were to use the unpooled t test with the conservative estimate for the degrees of

freedom. The t statistic for comparing the mean prices is 2.1. What can we say about the value of the p-value? A) p-value < 0.01 C) 0.05 < p-value < 0.10 B) 0.01 < p-value < 0.05 D) p-value > 0.10

Use the following to answer questions 19 and 20: Ten couples are participating in a small study on cholesterol. Neither the man nor the woman in each couple is known to have any problems with high cholesterol. The researcher conducting the study wishes to use the t test for matched pairs to determine if there is evidence that the cholesterol level for the husband tends to be higher than the cholesterol level for the wife. The cholesterol measurements for the ten couples are given below:

Couple 1 2 3 4 5 6 7 8 9 10 Husband’s cholesterol 224 310 266 332 244 178 280 276 242 260 Wife’s cholesterol 200 270 288 296 270 180 268 244 210 236

19. What are the hypotheses the researcher wishes to test?

A) Ho: D

µ = 0 versus Ha: D

µ > 0, where D

µ = the mean of the differences in cholesterol levels (Husband – Wife) for all couples without cholesterol problems.

B) Ho: p = ½ versus Ha: p ≠ ½, where p = the proportion of cholesterol levels of the husband that are higher than those of the wife.

C) Ho: p = ½ versus Ha: p < ½, where p = the proportion of cholesterol levels of the husband that are higher than those of the wife.

D) Ho: population median = 0 versus Ha: population median > 0, where the differences for which the median is calculated are measured as Husband – Wife.

20. What is the (approximate) value of the p-value?

A) 0.039 C) 0.079 B) 0.055 D) 0.172

Use the following to answer questions 21–24: A sportswriter wished to see if a football filled with helium travels farther, on average, than a football filled with air. To test this, the writer used 18 adult male volunteers. These volunteers were randomly divided into two groups of nine subjects each. Group 1 kicked a football filled with helium to the recommended pressure. Group 2 kicked a football filled with air to the recommended pressure. The mean yardage for

Group 1 was 1x = 30 yards with a standard deviation 1

s = 8 yards. The mean yardage for Group 2 was

2x = 26 yards with a standard deviation 2

s = 6 yards. Assume that the two groups of kicks are

independent. Let 1

µ and 2

µ represent the mean yardage we would observe for the entire population represented by the volunteers if all members of this population kicked, respectively, a helium-filled football

and an air-filled football. Let 1

σ and 2

σ be the corresponding population standard deviations.

21. Assuming two-sample t procedures are safe to use, what is a 99% confidence interval for 1

µ – 2

µ ? (Use the conservative value for the degrees of freedom.) A) 4 ± 4.7 yards C) 4 ± 7.7 yards B) 4 ± 6.2 yards D) 4 ± 11.2 yards

22. Suppose we wish to test the hypothesis that the groups are equivalent in how variable their kicks are.

To do this, we wish to test the hypotheses Ho: 1

µ = 2

µ versus Ha: 1

µ ≠ 2

µ . Assume the distribution of the lengths of these kicks is Normal. What can we say about the value of the p-value? A) p-value < 0.025 C) 0.05 < p-value < 0.10 B) 0.025 < p-value < 0.05 D) p-value > 0.10

23. Based on the confidence interval for the difference in means and the test for equality of the standard deviations, we can draw a conclusion about the distribution of the lengths of the kicks with the two different kinds of footballs. Determine which of the following statements is true. A) Both the means and standard deviations of the air and helium groups seem to be the same. B) The means of the air and helium groups seem to be the same. However, the standard deviations

seem to be different. C) The standard deviations of the air and helium groups seem to be the same. However, the means

seem to be different. D) Both the means and standard deviations of the air and helium groups seem to be different from

one another. 24. If we had used the more accurate software approximation to the degrees of freedom, what would be

the number of degrees of freedom for the two-sample t procedures? A) 8 C) 14 B) 9.374 D) 14.837

Use the following to answer questions 25–27: There is substantial interest in the health benefits of the consumption of high amounts of fiber in diets. A market research team is interested in the public acceptance of a new high-fiber cereal (more than 8 gm of fiber per serving) that is to be marketed. To that end, the researchers selected a random sample of subjects from one region of the country. The selected subjects were provided with two bowls of cereal. One bowl contained the new cereal and the other bowl a well-known and popular cereal. The bowls were presented in random order and subjects asked which cereal they preferred. The study was repeated independently in a second region. In region 1, of the 400 subjects, 220 preferred the new cereal; in region 2, 195 of the 300 subjects indicated a preference for the new cereal. 25. The researchers wanted to test whether the proportions of consumers who preferred the new high-

fiber cereal are the same or different in the two regions. What null and alternative hypotheses should they establish?

A) Ho: p1 = p2 against Ha: p1 > p2

B) Ho: p̂1 = p̂2 against Ha: p̂1 ≠ p̂2

C) Ho: p1 = p2 against Ha: p1 ≠ p2

D) Ho: p̂1 = p̂2 against Ha: p̂1 > p̂2

E) Ho: p1 − p2 = 0 against Ha: p1 − p2 = 0.5 26. What is the value of the test statistic?

A) t = −2.67 B) z = −2.70 C) z = −3.84 D) z = −2.66 E) t = −2.70

27. What is the p-value for this test?

A) 0.0077 B) 0.0035 C) 0.0070 D) 0.0038 E) < 0.0002

Use the following to answer questions 28 – 30: A study was conducted at the University of Waterloo on the impact characteristics of football helmets used in competitive high school programs. There were three types of helmets considered, classified according to liner type: suspension, padded-suspension, and padded. In the study, a measurement called the Gadd Severity Index (GSI) was obtained on each helmet using a standardized impact test. A helmet was deemed to have failed if the GSI was greater than 1200. Of the 81 helmets tested, 29 failed the GSI 1200 criterion. 28. Assume that the suspension helmets tested were selected at random. What are the point estimates

of the proportion of suspension helmets that fail and the standard error of the estimate, respectively? A) 0.36; 0.0028 B) 0.64; 0.053 C) 0.36; 0.053 D) 0.64; 0.0028 E) 0.36: 0.089

29. Based on the sample results, what is the 90% confidence interval estimate for the true population

proportion of suspension helmets that would fail the test? A) (0.304, 0.416) B) (0.256, 0.464) C) (0.213, 0.507) D) (0.272, 0.448) E) (0.553, 0.737)

30. If the test was to be conducted again, how many suspension-type helmets should be tested so that

the margin of error does not exceed 0.05 with 95% confidence? A) 355 B) 20 C) 271 D) 250 E) 82

Extra Credit 1. The scores on the Wechsler Intelligence Scale for Children (WISC) are thought to be Normally

distributed with a standard deviation of σ = 10. A simple random sample of 25 children is taken, and

each is given the WISC. The mean of the 25 scores is x = 104.32. Based on these data, what is a 95% confidence interval for µ ? A) 104.32 ± 0.78 B) 104.32 ± 3.29 C) 104.32 ± 3.92 D) 104.32 ± 19.60

2. A sample of size n = 27 is used to conduct a significance test for Ho: µ = 75 versus

Ha: µ > 75. The test statistic is t = 3.45. What are the degrees of freedom for this test statistic? A) 26 B) 27 C) 74 D) 75

3. A simple random sample of five female basketball players is selected. Their heights (in cm) are 170,

175, 169, 183, and 177. What is the standard error of the mean of these height measurements? A) 2.538 B) 2.837 C) 5.075 D) 5.675

Chs 9 – 12 For Questions 1 – 2 A sample of 50 male and 50 female infants were put on an experimental infant formula. After two weeks the parents of each infant were asked to fill out a questionnaire concerning infant satisfaction on the formula. One of the questions was “Did Baby Seem to Like the Formula?” with possible responses

1 – Like Very Much; 2 – Like Somewhat; 3 – Neutral; 4 – Dislike Somewhat; 5 – Dislike Very Much

The resulting data are presented below.

Gender Male Female Like Very Much 16 14 Like Somewhat 13 18 Neutral 17 16 Dislike Somewhat 3 2 Disike Very Much 1 0

1. This is a

a) 2 x 2 table. b) 2 x 5 table. c) 5 x 2 table. d) 5 x 5 table

2. The appropriate null hypothesis for this data is that

a) the distribution of parents’ responses on this question is the same for male and female infants. b) the distribution of gender is the same for each parents’ response to this question. c) gender and parents’ responses on this question are independent. d) gender and parents’ responses on this question are dependent

3. A study to compare two types of infant formula was run at two sites, one in Atlanta and the second in Denver. The study was run over a three-week period. Subjects at both sites were classified as dropouts if they left the study before the conclusion, or completers if they finished the study. The following table gives the number of dropouts and completers at each site. A chi-square test was performed and the result was X

2 = 5.101 with p-value = 0.024.

Responder Dropout Completer Atlanta 16 134 Denver 21 379

The correct conclusion is a) we found evidence to suggest that Atlanta had a greater dropout rate. b) we found evidence to suggest that Denver had a greater dropout rate. c) any differences can be explained by sampling variability. d) there is no association between responder and dropout rate.

For Questions 4 – 7 A study was performed to examine the personal goals of children in grades 4, 5, and 6. A random sample of students was selected for each of the grades from schools in Georgia. The students received a questionnaire regarding personal goals. They were asked what they would most like to do at school: make good grades, be popular, or be good at sports. Results are presented in the table below by the sex of the child.

Make good grades Be popular Be good in sports

Boys 96 32 94

Girls 295 45 40

4. The proportion of boys who chose the goal “be good in sports” and the proportion of girls who chose

the goal “be good in sports” are a) proportion of boys = 0.42, proportion of girls = 0.07. b) proportion of boys = 0.70, proportion of girls = 0.30. c) proportion of boys = 0.16, proportion of girls = 0.07. d) proportion of boys = 0.42, proportion of girls = 0.11.

5. Suppose we wish to test the null hypothesis that there are no differences among the proportion of boys

and the proportion of girls choosing each of the three personal goals. Under the null hypothesis, the expected number of boys that would select “be good in sports” is a) 49.4 b) 67 c) 74 d) 33

6.. Suppose we wish to test the null hypothesis that there are no differences among the proportion of

boys and the proportion of girls choosing each of the three personal goals. The value of the chi- square statistic X

2 is

a) 0.2893 b) 1.2644 c) 90.0266 d) 45.4335

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