Statistics – Test 4

A level 0.90 confidence interval is A) any interval with margin of error ± 0.90. B) an interval computed from sample data by a method that has probability 0.90 of producing an interval containing the true value of the parameter of interest. C) an interval with margin of error ± 0.90, which is also correct 90% of the time. D) an interval computed from sample data by a method that guarantees that the probability the interval computed contains the parameter of interest is 0.90.

An agricultural researcher plants 100 plots with a new variety of corn. The average yield for these plots is x = 150 bushels per acre. Assume that the yield per acre for the new variety of corn follows a normal distribution with unknown mean m and standard deviation s = 20 bushels per acre. A 90% confidence interval for m is A) 150 ± 2.00. B) 150 ± 3.29. C) 150 ± 3.92. D) 150 ± 32.90.

An agricultural researcher plants 100 plots with a new variety of corn. The average yield for these plots is x = 150 bushels per acre. Assume that the yield per acre for the new variety of corn follows a normal distribution with unknown mean m and standard deviation s = 20 bushels per acre. Which of the following would produce a confidence interval with a smaller margin of error than the 90% confidence interval you computed above? A) Plant only 25 plots rather than 100 because 25 are easier to manage and control. B) Plant 500 plots rather than 100. C) Compute a 99% confidence interval rather than a 90% confidence interval. The increase in confidence indicates that we have a better interval. D) None of the above.

You plan to construct a confidence interval for the mean m of a normal population with (known) standard deviation s. Which of the following will reduce the size of the margin of error? A) Use a lower level of confidence. B) Increase the sample size. C) Reduce s. D) All of the above.

Suppose that the population of the scores of all high school seniors that took the SAT-M (SAT math) test this year follows a normal distribution with mean m and standard deviation s = 100. You read a report that says, "On the basis of a simple random sample of 100 high school seniors that took the SAT-M test this year, a confidence interval for m is 512.00 ± 25.76." The confidence level for this interval is A) 90%. B) 95%. C) 99%. D) over 99.9%.

The heights of 15-year-old American boys, in inches, are normally distributed with mean m and standard deviation s = 2.4. I select a simple random sample of four 15-year-old American boys and measure their heights. The four heights, in inches, are 63 69 62 66. 6. Based on these data, a 99% confidence interval for m, in inches, is A) 65.00 ± 1.55. B) 65.00 ± 2.35. C) 65.00 ± 3.09. D) 65.00 ± 4.07.

The heights of 15-year-old American boys, in inches, are normally distributed with mean m and standard deviation s = 2.4. I select a simple random sample of four 15-year-old American boys and measure their heights. The four heights, in inches, are 63 69 62 66. If I wanted the margin of error for the 99% confidence interval to be ± 1 inch, I should select a simple random sample of size A) 2. B) 7. C) 16. D) 39.

The scores of a certain population on the Wechsler Intelligence Scale for Children (WISC) are thought to be normally distributed with mean m and standard deviation s = 10. A simple random sample of 25 children from this population is taken and each is given the WISC. The mean of the 25 scores is x = 104.32. Based on these data, a 95% confidence interval for m is A) 104.32 ± 0.78. B) 104.32 ± 3.29. C) 104.32 ± 3.92. D) 104.32 ± 19.60

Suppose a histogram of the 25 WISC scores is given below. Based on this histogram, we would conclude A) the 95% confidence interval for m computed from these data is very reliable. B) the 95% confidence interval for m computed from these data is not very reliable. C) the 95% confidence interval for m computed from these data is actually a 99% confidence interval.. D) the 95% confidence interval for m computed from these data is actually a 90% confidence interval.

Suppose we want a 90% confidence interval for the average amount spent on entertainment (movies, concerts, dates, etc.) by Freshman in their first semester at a large university. The interval is to have a margin of error of $2, and the amount spent has a normal distribution with a standard deviation s = $30. The number of observations required is closest to A) 25. B) 30. C) 609. D) 865.

In their advertisements, the manufacturers of a diet pill would like to claim that taken daily, their pill will produce a mean weight loss of more than 10 pounds in one month. In order to determine if this is a valid claim, they hire an independent testing agency, which then selects 25 people to take the pill daily for a month. The agency should be testing the null hypothesis H0: m = 10 and the alternative hypothesis A) Ha: m 10. C) Ha: m ¹ 10. D) Ha: m ¹ 10 ± SD/Square root n

Suppose we are testing the null hypothesis H0 : m = 20 and the alternative Ha: m ¹ 20, for a normal population with s = 6. A random sample of nine observations are drawn from the population and we find the sample mean of these observations is x = 17. The P-value is closest to A) 0.0668. B) 0.1336. C) .0332. D) .3085.

Is the mean height for all adult American males between the ages of 18 and 21 now over 6 feet? If the population of all adult American males between the ages of 18 and 21 has mean height of m feet and standard deviation s feet, to answer this question one would test which of the following null and alternative hypotheses? A) H0: m = 6 vs. Ha: m > 6. B) H0: m = 6 vs. Ha: m < 6. C) H0: m = 6 vs. Ha: m ¹ 6. D) H0: m = 6 vs. Ha: m = 6 ± x , assuming our sample size is n.

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 of 115 and standard deviation s = 25. You suspect that incoming freshman have a mean m, which is different from 115, because they are often excited yet anxious about entering college. To test your suspicion, you test the hypotheses H0: m = 115, Ha: m ¹ 115. You give the SSHA to 25 students who are incoming freshman and find their mean score. Assuming that the scores of all incoming freshmen are approximately normal with the same standard deviation as the scores of all college students, the P-value of the test of the null hypothesis is A) the probability, assuming the null hypothesis is true, that the test statistic will take a value at least as extreme as that actually observed. B) the probability, assuming the null hypothesis is false, that the test statistic will take a value at least as extreme as that actually observed. C) the probability the null hypothesis is true. D) the probability the null hypothesis is false.

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 of 115 and standard deviation s = 25. You suspect that incoming freshman have a mean m, which is different from 115, because they are often excited yet anxious about entering college. To test your suspicion, you test the hypotheses H0: m = 115, Ha: m ¹ 115. In testing these hypotheses, which of the following would be strong evidence against the null hypothesis? A) using a small level of significance B) using a large level of significance C) obtaining data with a small P-value D) obtaining data with a large P-value

A university administrator obtains a sample of the academic records of past and present scholarship athletes at the university. The administrator reports that no significant difference was found in the mean GPA (grade point average) for male and female scholarship athletes (P = 0.287). This means A) the GPAs for male and female scholarship athletes are identical except for 28.7% of the athletes. B) the maximum difference in GPAs between male and female scholarship athletes is 0.287. C) the chance of obtaining a difference in GPAs between male and female scholarship athletes as large as that observed in the sample if there is no difference in mean GPAs is 0.287. D) the chance that a pair of randomly chosen male and female scholarship athletes would have a significant difference in GPAs is 0.287.

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 s = 25. You suspect that incoming freshman have a mean m which is different from 115 because they are often excited yet anxious about entering college. To test your suspicion, you test the hypotheses H0: m = 115, Ha: m ¹ 115. You give the SSHA to 25 students who are incoming freshman and find their mean score is 116.2. Assuming that the scores of all incoming freshmen are approximately normal with the same standard deviation as the scores of all college students, the Pvalue of your test is A) 0.1151. B) 0.2302. C) 0.4052. D) 0.8104.

The nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) m and standard deviation s = 0.1. The brand advertises that the mean nicotine content of their cigarettes is 1.5, but measurements on a random sample of 400 cigarettes of this brand gave a mean of x = 1.52. Is this evidence that the mean nicotine content is actually higher than advertised? To answer this, test the hypotheses H0: m = 1.5, Ha: m > 1.5 at significance level a = 0.01. You conclude A) that H0 should be rejected. B) that H0 should not be rejected. C) that Ha should be rejected. D) that there is a 5% chance that the null hypothesis is true.

A certain population follows a normal distribution with mean m and standard deviation s = 2.5. You collect data and test the hypotheses H0: m = 1, Ha: m ¹ 1. You obtain a P-value of 0.022. Which of the following is true? A) A 95% confidence interval for m will include the value 1. B) A 95% confidence interval for m will include the value 0. C) A 99% confidence interval for m will include the value 1. D) A 99% confidence interval for m will include the value 0.

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.

A researcher wishes to determine if students are able to complete a certain puzzle more quickly while exposed to a pleasant floral scent. Suppose the time (in seconds) needed for high school students to complete the puzzle while exposed to the scent follows a normal distribution with mean m and standard deviation s = 4. Suppose, also, that in the general population of all high school students, the time needed to complete the puzzle follows a normal distribution with mean 80 and standard deviation s = 4. The researcher, therefore, decides to test the hypotheses H0: m = 80, Ha: m < 80. To do so, the researcher has 10,000 high school students complete the puzzle in the presence of the floral scent. The mean time for these students is x = 80.2 seconds and the P-value is less than 0.0001. It is appropriate to conclude which of the following? A) The researcher has proven that for high school students, a pleasant floral scent substantially improves the time it takes to complete the puzzle. B) The researcher has strong evidence that for high school students, a pleasant floral scent improves the time it takes to complete the puzzle. C) The researcher has moderate evidence that for high school students, a pleasant floral scent substantially improves the time it takes to complete the puzzle. D) None of the above.

Suppose that two high school students decide to see if they get the same results as the researcher. They both do the puzzle while in the presence of the pleasant floral scent. The mean of their times is x = 80.2 seconds, the same as that of the researcher. It is appropriate to conclude which of the following? A) They have reproduced the results of the researcher and their P-value will be the same as that of the researcher. B) They have reproduced the results of the researcher, but their P-value will be slightly smaller than that of the researcher. C) They will reach the same statistical conclusion as the researcher, but their P-value will be a bit different than that of the researcher. D) None of the above.

A medical researcher is working on a new treatment for a certain type of cancer. The average survival time after diagnosis on the standard treatment is two years. In an early trial, she tries the new treatment on three subjects who have an average survival time after diagnosis of four years. Although the survival time has doubled, the results are not statistically significant even at the 0.10 significance level. The explanation is A) the placebo effect is present, which limits statistical significance. B) the sample size is small. C) that although the survival time has doubled, in reality the actual increase is really two years. D) the calculation was in error. The researchers forgot to include the sample size.

An engineer designs an improved light bulb. The previous design had an average lifetime of 1200 hours. The new bulb had a lifetime of 1200.2 hours, using a sample of 40,000 bulbs. Although the difference is quite small, the effect was statistically significant. The explanation is A) that new designs typically have more variability than standard designs. B) that the sample size is very large. C) that the mean of 1200 is large. D) all of the above.

The nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) m and standard deviation s = 0.1. The brand advertises that the mean nicotine content of their cigarettes is 1.5, but you believe that the mean nicotine content is actually higher than advertised. To explore this, you test the hypotheses H0: m = 1.5, Ha: m > 1.5 and you obtain a P-value of 0.052. Which of the following is true? A) At the a = 0.05 significance level, you have proven that H0 is true. B) You have failed to obtain any evidence for Ha. C) There is some evidence against H0, and a study using a larger sample size may be worthwhile. D) This should be viewed as a pilot study and the data suggests that further investigation of the hypotheses will not be fruitful at the a = 0.05 significance level.

In assessing the validity of any test of hypotheses, it is good practice to A) examine the probability model that serves as a basis for the test by using exploratory data analysis on the data. B) determine exactly how the study was conducted. C) determine what assumptions the researchers made. D) all of the above.

A radio show runs a phone-in survey each morning. One morning the show asked its listeners whether they would prefer Congress or the President to set policy for the nation. The majority of those phoning in their responses answered "Congress," and the station reported the results as statistically significant. We may safely conclude A) that there is deep discontent in the nation with the President. B) that it is unlikely that if all Americans were asked their opinion, that the result would differ from that obtained in the poll. C) that there is strong evidence that the majority of Americans prefer Congress to set national policy. D) that very few, other than the majority of those phoning in their responses, preferred Congress to set policy for the nation.

Does 30 minutes of aerobic exercise each day provide significant improvement in mental performance? To investigate this issue, a researcher conducted a study with 150 adult subjects who performed aerobic exercise each day for a period of six months. At the end of the study, 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults. Nine of these variables were significantly better (in the sense of statistical significance) at the a = 0.05 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole, and one variable was significantly better at the a = 0.01 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole. It would be correct to conclude A) that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance. B) that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides improvement for the variable that was significant at the a = 0.01 level. We should be somewhat cautious about making claims for the variables that were significant at the a = 0.05 level. C) These results would have provided very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance if the number of subjects had been larger. It is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured. D) none of the above.

Which of the following is an example of a matched pairs design? A) A teacher compares the pre-test and post-test scores of students. B) A teacher compares the scores of students using a computer-based method of instruction with the scores of other students using a traditional method of instruction. C) A teacher compares the scores of students in her class on a standardized test with the national average score. D) A teacher calculates the average of scores of students on a pair of tests and wishes to see if this average is larger than 80%.

To estimate m, the mean salary of full professors at American colleges and universities, you obtain the salaries of a random sample of 400 full professors. The sample mean is x = $73,220 and the sample standard deviation is s = $4400. A 99% confidence interval for m is A) 73220 ± 11440. B) 73220 ± 572. C) 73220 ± 431. D) 73220 ± 28.6.

You are thinking of using a t-procedure to test hypotheses about the mean of a population using a significance level of 0.05. You suspect the distribution of the population is not normal and may be moderately skewed. Which of the following statements is correct? A) You should not use the t-procedure because the population does not have a normal distribution. B) You may use the t-procedure provided your sample size is large, say at least 50. C) You may use the t-procedure, but you should probably claim the significance level is only 0.10. D) You may not use the t-procedure. t-procedures are robust to nonnormality for confidence intervals but not for tests of hypotheses.

A SRS of 20 third grade children is selected in Chicago and each is given a test to measure his or her reading ability. In the sample, the mean score is 64 points and the standard deviation is 12 points. The standard error of the mean is A) 14.31. B) 7.20. C) 2.68. D) 0.60.

A SRS of 20 third grade children is selected in Chicago and each is given a test to measure his or her reading ability. In the sample, the mean score is 64 points and the standard deviation is 12 points. We are interested in a 95% confidence interval for the population mean score. The margin of error associated with the confidence interval is A) 2.68 points. B) 4.64 points. C) 5.62 points. D) 6.84 points.

A SRS of 20 third grade children is selected in Chicago and each is given a test to measure his or her reading ability. In the sample, the mean score is 64 points and the standard deviation is 12 points. A 90% confidence interval for the population mean score based on these data is A) 64 ± 2.68 points. B) 64 ± 4.64 points. C) 64 ± 5.62 points. D) 64 ± 6.84 points.

The one-sample t statistic from a sample of n = 19 observations for the two-sided test of H0: m = 6, Ha: m ¹ 6 has the value t = 1.93. Based on this information A) we would reject the null hypothesis at a = 0.10. B) 0.025 < P-value < 0.05. C) we would reject the null hypothesis at a = 0.05. D) both b) and c) are correct.

Bags of a certain brand of tortilla chips claim to have a net weight of 14 ounces. Net weights actually vary slightly from bag to bag and are normally distributed with mean m. A representative of a consumer advocate group wishes to see if there is any evidence that the mean net weight is less than advertised and so intends to test the hypotheses H0: m = 14, Ha: m < 14. To do this, he selects 16 bags of this brand at random and determines the net weight of each. He finds the sample mean to be x = 13.88 and the sample standard deviation to be s = 0.24. Based on these data, A) we would reject H0 at significance level 0.10 but not at 0.05. B) we would reject H0 at significance level 0.05 but not at 0.025. C) we would reject H0 at significance level 0.025 but not at 0.01. D) we would reject H0 at significance level 0.01.

Bags of a certain brand of tortilla chips claim to have a net weight of 14 ounces. Net weights actually vary slightly from bag to bag and are normally distributed with mean m. A representative of a consumer advocate group wishes to see if there is any evidence that the mean net weight is less than advertised and so intends to test the hypotheses H0: m = 14, Ha: m < 14. To do this, he selects 16 bags of this brand at random and determines the net weight of each. He finds the sample mean to be x = 13.88 and the sample standard deviation to be s = 0.24. Suppose we were not sure if the distribution of net weights was normal. In which of the following circumstances would we not be safe using a t procedure in this problem? A) The mean and median of the data are nearly equal. B) A histogram of the data shows moderate skewness. C) A stemplot of the data has a large outlier. D) The sample standard deviation is large.

An SRS of 100 laborers who use the services of a national temporary employment agency found that in the past year the average number of days worked by these laborers was x = 107 days, with standard deviation s = 45 days. Assume the distribution of the number of days worked in the population of laborers using this employment agency is approximately normal, with mean m. Are these data evidence that m has lowered from the value of 120 days of 5 years ago? To determine this, we test the hypotheses H0: m = 120, Ha: m < 120 The appropriate degrees of freedom for this test are A) 45. B) 99. C) 100. D) 120.

An SRS of 100 laborers who use the services of a national temporary employment agency found that in the past year the average number of days worked by these laborers was x = 107 days, with standard deviation s = 45 days. Assume the distribution of the number of days worked in the population of laborers using this employment agency is approximately normal, with mean m. Are these data evidence that m has lowered from the value of 120 days of 5 years ago? To determine this, we test the hypotheses H0: m = 120, Ha: m < 120 Based on the data, the value of the one-sample t statistic is A) 3.41. B) 2.89. C) 2.67. D) 2.38.

An SRS of 100 laborers who use the services of a national temporary employment agency found that in the past year the average number of days worked by these laborers was x = 107 days, with standard deviation s = 45 days. Assume the distribution of the number of days worked in the population of laborers using this employment agency is approximately normal, with mean m. Are these data evidence that m has lowered from the value of 120 days of 5 years ago? To determine this, we test the hypotheses H0: m = 120, Ha: m < 120 The P-value for the one-sample t test is A) larger than 0.10. B) between 0.10 and 0.05. C) between 0.05 and 0.01. D) below 0.01.

Suppose the mean and standard deviation obtained were based on a sample of 25 laborers rather than 100. The P-value would be A) larger. B) smaller. C) unchanged because the difference between x and the hypothesized value m = 120 is unchanged. D) unchanged because the variability measured by the standard deviation stays the same.

A 95% confidence interval for the population mean number of days m that laborers using the services of this agency have worked during the past year is A) 107 ± 45. B) 107 ± 8.9. C) 107 ± 7.5. D) 107 ± 4.5.

A certain population follows a normal distribution with mean m and standard deviation s = 1.2. You construct a 95% confidence interval for m and find it to be 1.1 ± 0.8. Which of the following is true? A) A test of the hypotheses H0: m = 1.2, Ha: m ¹ 1.2 would be rejected at the 0.05 level. B) A test of the hypotheses H0: m = 1.1, Ha: m ¹ 1.1 would be rejected at the 0.05 level. C) A test of the hypotheses H0: m = 0, Ha: m ¹ 0 would be rejected at the 0.05 level. D) All the above.