Name: Date: 1. Suppose you wished to see if there is evidence that the psychic is doing better than just guessing. To do this you test the hypotheses

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1 Statistics Semester Two Review Name: Date: Use the following to answer questions 1-2. A noted psychic was tested for ESP. The psychic was presented with 200 cards face down and asked to determine if the card was one of five symbols: a star, cross, circle, square or three wavy lines. The psychic was correct in 50 cases. Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial. 1. Suppose you wished to see if there is evidence that the psychic is doing better than just guessing. To do this you test the hypotheses H 0 : p = 0.20, H a : p > The P-value of your test is A) between.10 and.05. B) between.05 and.01. C) between.01 and.001. D) below Based on the results of the test, a 95% confidence interval for p is (assuming that the 200 trials can be treated as an SRS from the population of all guesses the psychic would make in his lifetime) A) 0.25 ± B) 0.25 ± C) 0.25 ± D) We can assert that p = 0.20 with 100% confidence because the psychic is just guessing. Page 1

2 Use the following to answer questions 3-4. When a police officer responds to a call for help in a case of spousal abuse, what should the officer do? A randomized controlled experiment in Charlotte, North Carolina studied three police responses to spousal abuse: advise and possibly separate the couple, issue a citation to the accused, and arrest the accused. The effectiveness of the three responses was determined by re-arrest rates. The table following shows these rates. Assigned Treatment Number of subsequent arrests Arrest Citation Advise/Separate This is an r c table. The number r has value A) 2. B) 3. C) 4. D) Suppose we wish to test the null hypothesis that the proportion of subsequent arrests is the same, regardless of the treatment assigned. Under the null hypothesis, the expected number of times no subsequent arrest would occur for the treatment Advise/Separate is A) 177. B) 181. C) 187. D) 543. Page 2

3 Use the following to answer questions 5-7. To study the export activity of manufacturing firms in Korea, questionnaires were mailed to a SRS of firms in each of four industries that export many of their products. The response rate was low, and to compare the industries it is important that the response rates from the different industries be similar. Do the data in the Minitab output shown in the following table provide evidence of a difference in response rate between the four industries? The output includes the cell counts, the expected cell counts, and the chi-square statistic. Expected counts are printed below observed counts. 5. This is an r c table. The number r has value A) 1. B) 2. C) 3. D) 4. Page 3

4 6. The P-value for testing the null hypothesis that the probability of response is the same for the four industries against the alternative that the probability is different for the four industries A) is greater than B) is between and C) is between and D) cannot be determined because these are not the hypotheses being tested by the chisquare test. 7. The appropriate degrees of freedom for the chi-square statistic is A) 1. B) 2. C) 3. D) 4. Use the following to answer question 8. A study examines the personal goals of children in grades 4, 5, and 6. A random sample of students was selected for each of the grades 4, 5, and 6 from schools in Georgia. The students received a questionnaire regarding achievement of personal goals. They were asked what they would most like to do at school: make good grades, be good at sports, or be popular. Results are presented in the following table by the sex of the child. Boys Girls Make good grades Be popular Be good in sports The numerical value of the chi-square statistic for this table is A) B) C) D) Page 4

5 9. To assess the accuracy of a laboratory scale, a standard weight known to weigh 1 gram is repeatedly weighed a total of n times, and the mean of the weighings is computed. Suppose the scale readings are normally distributed, with unknown mean m and standard deviation σ = 0.01 g. How large should n be, so that a 95% confidence interval for m has a margin of error of ± ? A) 100 B) 196 C) 10,000 D) 38,416 Use the following to answer question 10. A study examines the personal goals of children in grades 4, 5, and 6. A random sample of students was selected for each of the grades 4, 5, and 6 from schools in Georgia. The students received a questionnaire regarding achievement of personal goals. They were asked what they would most like to do at school: make good grades, be good at sports, or be popular. Results are presented in the following table by the sex of the child. Boys Girls Make good grades Be popular Be good in sports Which hypotheses are being tested by the chi-square test? A) The null hypothesis is that personal goals and gender are independent, and the alternative is that they are dependent. B) The null hypothesis is that the mean personal goal is the same for boys and girls, and the alternative is that the means differ. C) The distribution of personal goals is different for boys and girls. D) The distribution of gender is different for the three different personal goals. Page 5

6 Use the following to answer questions A food company is developing a new granola bar, and its market analysts are currently working on preliminary studies of the packaging design. To help with a marketing strategy, the company was first interested in whether the appeal of the packaging design for the new product (and hence the appeal of the product itself) was related to a person's gender. There were 100 male and 100 female volunteers available for purposes of evaluation. Both males and females rated the design on a scale of 1 to 10, 1 being very unappealing and 10 being very appealing. The mean rating for males was = 7.4, with a standard deviation s 1 = 1.5. The mean rating for females was = 8.0, with a standard deviation s 2 = 2.0. Let µ 1 and µ 2 represent the mean ratings we would observe for the populations of males and females, respectively, and assume our samples can be regarded as samples from these populations. 11. A 90% confidence interval for µ 1 µ 2 is (use the conservative value for the degrees of freedom) A) 0.6 ± B) 0.6 ± C) 0.6 ± D) 0.6 ± Suppose the researcher had wished to test the hypotheses H 0 : m 1 = m 2, H a : m 1! m 2. The P-value for the test is (use the conservative value for the degrees of freedom) A) larger than B) between 0.10 and C) between 0.05 and D) below If we had used the more accurate software approximation to the degrees of freedom, we would have used which of the following for the number of degrees of freedom for the t procedures? A) 198 B) 191 C) 184 D) 99 Page 6

7 14. Suppose the researcher had wished to test the hypotheses H 0 : m 1 = m 2, H a : m 1! m 2. The numerical value of the two-sample t statistic is A) B) C) D) Use the following to answer question 15. The American Veterinary Medical Association conducted a survey of veterinary clinics to estimate the proportion that do not treat large animals (cows, horses, etc.). The survey was mailed to a random sample of 120 veterinary clinics throughout the country and of these, 88 responded that they do not treat large animals. 15. The standard error of the proportion of clinics that do not treat large animals is A) B) C) D) Is the mean age at which American children can first read now under 4 years? If the population of all American children has mean age of µ years until they begin to read and standard deviation σ years, one would test which of the following null and alternative hypotheses to answer this question? A) H 0 : µ = 4 vs. H a : µ > 4 B) H 0 : µ = 4 vs. H a : µ < 4 C) H 0 : µ = 4 vs. H a : µ 4 D) H 0 : µ = 4 vs. H a : µ = 4 ±, assuming our sample size is n Page 7

8 Use the following to answer questions A researcher believes that college students spend less time participating in extra-curricular activites than they did ten years ago. An SRS of 100 college students found that in the past year the average number of hours spent per semester in extracurricular activities was = 107 hours, with standard deviation s = 45 hours. Assume the distribution of the number of hours spent by college students per semester participating in extracurricular activities is approximately normal, with mean µ. Are these data evidence that µ has lowered from the value of 120 hours of ten years ago? To determine this, we test the hypotheses H 0 : µ = 120, H a : µ < The P-value for the one-sample t test is A) larger than B) between 0.10 and C) between 0.05 and D) below The appropriate degrees of freedom for this test are A) 45. B) 99. C) 100. D) Based on the data, the value of the one-sample t statistic is A) B) C) D) In formulating hypotheses for a statistical test of significance, the null hypothesis is often A) a statement of no effect or no difference. B) the probability of observing the data you actually obtained. C) a statement that the data are all 0. D) Page 8

9 Use the following to answer questions A random sample of 1400 Internet users was selected from the records of a large Internet provider and asked whether they would use the Internet or the library to obtain information about health issues. Of these, 872 said they would use the Internet. 21. If the Internet provider wanted an estimate of the proportion that would use the Internet rather than the library, with a margin of error of at most 0.01 in a 99% confidence interval, how large a sample size would be required? A) 1400 B) 8432 C) 10,000 D) 16, The proportion that would use the Internet rather than the library is A) B) C) D) The standard error of the proportion that would use the Internet rather than the library is A) B) C) D) Suppose we want a 90% confidence interval for the average amount of time (in minutes) spent per week on homework by the students in a large introductory statistics course at a large university. The interval is to have a margin of error of 2 minutes, and the amount spent has a normal distribution with a standard deviation σ = 30 minutes. The number of observations required is closest to A) 25. B) 30. C) 609. D) 865. Page 9

10 Use the following to answer questions A SRS of 20 commuters to the Los Angeles metropolitan area are selected, and each is asked how far they commute to work each day. In the sample, the mean distance is 64 miles and the standard deviation is 12 miles. Assume that in the population of all commuters to the Los Angeles metropolitan area, daily commuting distance follows a normal distribution, with some mean µ. 25. The standard error of the mean is A) B) C) D) We are interested in a 95% confidence interval for the population mean commuting distance. The margin of error associated with the confidence interval is A) 2.68 miles. B) 4.64 miles. C) 5.62 miles. D) 6.84 miles. 27. A 90% confidence interval for the population mean commuting distance based on these data is A) 64 ± 2.68 miles. B) 64 ± 4.64 miles. C) 64 ± 5.62 miles. D) 64 ± 6.84 miles. Page 10

11 Use the following to answer questions A large Midwestern University studied student performance in a statistics course required by its mechanical engineering majors. Of interest is the time spent in extracurricular activities and whether a student earned a B or better in the course. Extracurricular Activities (hours per week) Grade <2 2 to 12 >12 B or better Below B This is an r c table. The number r has value A) 2. B) 3. C) 4. D) Suppose we wish to test the null hypothesis that there is no association between success in the course and the time spent in extracurricular activities. Under the null hypothesis, the expected number of successful students (B or better) who engage in < 2 hours of extracurricular activities is A) B) C) D) An SRS of 25 male faculty members at a large university found that 10 felt that the university was supportive of female and minority faculty. An independent SRS of 20 female faculty found that 5 felt that the university was supportive of female and minority faculty. Let p 1 and p 2 represent the proportion of all male and female faculty, respectively, at the university who felt that the university was supportive of female and minority faculty at the time of the survey. A 95% plus four confidence interval for p 1 p 2 is A) ± B) ± C) 0.15 ± D) 0.15 ± Page 11

12 Use the following to answer question 31. A newspaper conducted a statewide survey concerning a proposal to raise taxes to prevent budget cuts to education. The newspaper took a random sample (assume it is an SRS) of 1200 registered voters and found that 580 would vote to raise taxes. Let p represent the proportion of registered voters in the state that would vote to raise taxes. 31. A 90% confidence interval for p is A) ± B) ± C) ± D) ± Use the following to answer question 32. A TV news program conducts a call-in-poll about a proposed city ban on smoking in public places. Of the 2467 callers, 1900 were opposed to the ban. 32. Let p be the proportion of citizens in the city opposed to the ban on smoking in public places. To determine if the majority of citizens in the city oppose the ban on smoking in public places, A) test the hypotheses H 0 : p = 0.50, H a : p B) test the hypotheses H 0 : p = 0.50, H a : p > C) test the hypotheses H 0 : p = 0.77, H a : p > D) none of the above. 33. Is there evidence that the proportion of freshmen who graduated in the bottom third of their high school class in 2001 was reduced because of the tougher admission standards adopted in 2000, compared to the proportion in 1999? To determine this, you test the hypotheses The P-value of your test is A) between.10 and.05. B) between.05 and.01. C) between.01 and.001. D) below.001. H 0 : p 1 = p 2, H a : p 1 > p 2. Page 12

13 Use the following to answer question 34. A random-digit dialing sample of 1750 adults found that 1140 used the Internet. Of those who used the Internet, 940 said they expect businesses to have Web sites containing product information; 320 of the 610 nonusers said this too. 34. Let and be the proportion of all parts users and nonusers, respectively, that expect businesses to have Web sites containing product information. A 90% confidence interval for is A) B) C) D) A Type II error is A) rejecting the null hypothesis when it is true. B) accepting the null hypothesis when it is false. C) incorrectly specifying the null hypothesis. D) incorrectly specifying the alternative hypothesis. Use the following to answer question 36. A newspaper conducted a statewide survey concerning a proposal to raise taxes to prevent budget cuts to education. The newspaper took a random sample (assume it is an SRS) of 1200 registered voters and found that 580 would vote to raise taxes. Let p represent the proportion of registered voters in the state that would vote to raise taxes. 36. How large a sample n would you need to estimate p with margin of error 0.01 with 95% confidence? Use the guess p = 0.5 as the value for p. A) n = 49 B) n = 1500 C) n = 4800 D) n = 9604 Page 13

14 Use the following to answer question 37. In a large midwestern university (the class of entering freshmen is 6000 or more students), an SRS of 100 entering freshmen in 1999 found that 20 finished in the bottom third of their high school class. Admission standards at the university were tightened in In 2001, an SRS of 100 entering freshmen found that 10 finished in the bottom third of their high school class. Let and be the proportion of all entering freshmen in 1999 and 2001, respectively, who graduated in the bottom third of their high school class. 37. Is there evidence that the proportion of freshmen who graduated in the bottom third of their high school class in 2001 has been reduced, as a result of the tougher admission standards adopted in 2000, compared to the proportion in 1999? To determine this, you test the hypotheses H 0 : p 1 = p 2, H a : p 1 > p 2. The value of the z-statistic for testing these hypotheses is A) z = B) z = C) z = D) z = Use the following to answer question 38. We wish to see if, on average, traffic is moving at the posted speed limit of 65 miles per hour along a certain stretch of Interstate 70. On each of four randomly selected days, a randomly selected car is timed and the speed of the car is recorded. The observed speeds are 70, 65, 70, and 75 miles per hour. Assuming that speeds are normally distributed with mean µ, we test whether, on average, traffic is moving at 65 miles per hour, by testing the hypotheses H 0 : µ = 65, H a : µ Based on the data, the value of the one-sample t statistic is A) 5. B) C) D) Page 14

15 Use the following to answer questions A scientist was curious if younger couples with at least one child were becoming more prevalent in urban areas. He decided to see if the mean age of the older spouse of couples with at least one child living within the city limits of a large metropolitan area was lower than that of the elder spouse of couples with at least one child living outside the city limits. He selected an SRS of 100 couples with at least one child living within the city limits and determined the mean age to be = 39 years, with a standard deviation s 1 = 8 years. He also selected an independent SRS of 100 couples with at least one child and determined the mean age to be = 40 years, with a standard deviation s 2 = 10 years. Let µ 1 and µ 2 represent the mean ages of the populations of all couples with at least one child living inside and outside the city limits, respectively. Suppose the researcher is interested in testing the hypotheses H 0 : µ 1 = µ 2, H a : µ 1 < µ The numerical value of the t statistic is A) B) C) D) The P-value for the test is (use the conservative value for the degrees of freedom) A) larger than B) between 0.10 and C) between 0.05 and D) below Page 15

16 Use the following to answer questions Do women and men treat female and male children differently? An observational study was conducted near the primate exhibit at the Columbus Zoo on weekends in The data are from 39 groups of three one adult female, one adult male, and one toddler, in which the toddler was being carried. Recorded below is which adult (male of female) was carrying the toddler by the sex of the toddler. Sex of Toddler Sex of adult carrying toddler Male Female Male 8 17 Female Suppose we wish to test the null hypothesis that the proportion of male adults and the proportion of female adults carrying a toddler is the same, regardless of the sex of the toddler. Under the null hypothesis, the expected number of times a male adult would be carrying a male toddler according to the table is A) 9. B) 8. C) 6. D) The numerical value of the chi-square statistic for testing independence of sex of the adult of sex of the toddler is A) B) C) D) If we reject the null hypothesis when, in fact, it is true, we have A) committed a Type I error. B) committed a Type II error. C) a probability of being correct that is equal to the P-value. D) a probability of being correct that is equal to 1 P-value. Page 16

17 Use the following to answer questions Recent revenue shortfalls in a Southern state led to a reduction in the state budget for higher education. To offset the reduction, the largest state university proposed a 25% tuition increase. It was determined that such an increase was needed to simply compensate for the lost support from the state. Random samples of 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors from the university were asked whether they were strongly opposed to the increase, given that it was the minimum increase necessary to maintain the university's budget at current levels. The results are given in the following table. Year Strongly Opposed Freshman Sophomore Junior Senior Yes No Suppose we wish to test the null hypothesis that there is no association between their year in school and students' opinion. Under the null hypothesis, what is the expected number of strongly opposed seniors? A) 18 B) 25 C) 30.5 D) The Chi-square statistic for this data equals 219. The P-value is A) greater than.10. B) between 0.05 and C) between 0.01 and D) less than The appraised values of three recently sold houses in the Columbus area are (in thousands of dollars) 260, 315, and 295. The standard error of the mean of these three appraised values is A) B) C) D) Page 17

18 Use the following to answer questions The housing commission of Ashland County is interested in finding out more about the number of rental units that qualify as low-income housing but do not meet the minimum standard living requirements in Van Township and Marlin Township. Units are randomly selected in both townships. Of the 85 low-income units sampled in Van Township (Township 1), 17 do not meet minimum requirements. Of the 80 units sampled in Marlin Township (Township 2), 24 do not meet minimum requirements. 47. The value of the z-statistic for testing equality of the proportion of low-income rental units that do not meet minimum standards in the two townships is A) z = B) z = C) z = D) z = The hypotheses to test that the proportions of low-income rental units that fail to meet minimum requirements in the two townships are equal or not is A) B) C) D) 49. The mean area µ of the several thousand apartments in a new development by a certain builder is advertised to be 1250 square feet. A tenant group thinks this is inaccurate, because it is based on the square footage of apartments in an older development by the same builder. The group hires an engineer to measure a sample of apartments to verify its suspicion. The appropriate null and alternative hypotheses, H 0 and H a, for µ are A) H 0 : µ = 1250 and H a : µ B) H 0 : µ = 1250 and H a : µ < C) H 0 : µ = 1250 and H a : µ > D) cannot be specified without knowing the size of the sample used by the engineer. Page 18

19 50. Scores on the SAT Mathematics test (SAT-M) are believed to be normally distributed, with mean µ. The scores of a random sample of three students who recently took the exam are 550, 620, and 480. A 95% confidence interval for µ based on these data is A) ± B) ± C) ± D) ± Questions 51 to 55 are BONUS! 51. Suppose the time it takes college students to complete their degree follows a normal distribution, with mean µ and standard deviation σ = 1.2 years. Based on random sample of college students, you construct a 95% confidence interval for µ and find it to be 1.15 ± Which of the following is true? A) A test of the hypotheses H 0 : µ = 1.2, H a : µ 1.2 would be rejected at the 0.05 level. B) A test of the hypotheses H 0 : µ = 1.1, H a : µ 1.1 would be rejected at the 0.05 level. C) A test of the hypotheses H 0 : µ = 0, H a : µ 0 would be rejected at the 0.05 level. D) All the above. 52. A level α two-sided significance test rejects the null hypothesis H 0 :µ = µ 0 exactly when A) the z test statistic differs from µ 0. B) the value µ 0 falls outside a level 1-α confidence interval for µ. C) the z test statistic differs from µ. D) the value µ falls outside a level 1-α confidence interval for µ In testing hypotheses, if the consequences of rejecting the null hypothesis are very serious, we should A) use a very large level of significance. B) use a very small level of significance. C) insist that the P-value be smaller than the level of significance. D) insist that the level of significance be smaller than the P-value. Page 19

20 54. Suppose I had measured the lifetimes of a random sample of 100 tires rather than 25. Which of the following statements is true? A) The margin of error for our 95% confidence interval would increase. B) The margin of error for our 95% confidence interval would decrease. C) The margin of error for our 95% confidence interval would stay the same, since the level of confidence has not changed. D) σ would decrease. Use the following to answer question 55. A large Midwestern University studied student performance in a statistics course required by its mechanical engineering majors. Of interest is the time spent in extracurricular activities and whether a student earned a B or better in the course. Extracurricular Activities (hours per week) Grade <2 2 to 12 >12 B or better Below B Suppose we wish to display in a graphic the proportion of time successful students spend in the three levels of extracurricular activities. Which of the following graphical displays is best suited to this purpose? A) a pie chart B) a histogram C) a boxplot D) a stemplot Page 20