Lecture Prep 02: Type I & Type II Errors
Is the following statement true or false? If false, explain briefly. The alpha level depends on the sample size. A) The statement is false because the alpha level is set independently and does not depend on the sample size. B) The statement is false because the alpha level depends on the p-value, not the sample size. C) The statement is false because the alpha level depends on the test statistic, not the sample size. D) The statement is true.
A) The statement is false because the alpha level is set independently and does not depend on the sample size.
(c) A clean air standard requires that vehicle exhaust emissions not exceed specified limits for various pollutants. Many states require that cars be tested annually to be sure they meet these standards. Suppose state regulators double-check a random sample of cars that a suspect repair shop has certified as okay. They will revoke the shop's license if they find significant evidence that the shop is certifying vehicles that do not meet standards. Which type of error would the shop's owner consider more serious? A) Type I error B) Type II error
A) Type I error
Is the following statement true or false? If false, explain briefly. If we use an alpha level of 0.01, a p-value of 0.001 is statistically significant. A) True. B) False. If we use an alpha level of 0.01, then a z-score of 0.001 is significant. C) False. An alpha level of 0.01 gives n significance to a p-value of 0.001. D) False. A p-value of 0.001 would not be significant since it is less than alpha.
A) True.
(b) Before lending someone money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the applicant, totaling points they award for the person's income level, credit history, current debt burden, and so on. The higher the point total, the more convinced the bank is that it's safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. We can think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the person's score falls below the minimum cutoff will the bank reject the null and deny the loan. Which kind of error is it when the bank misses an opportunity to make a loan to someone who would have repaid it? A) Type I error B) Type II error
A) Type I error
(a) Before lending someone money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the applicant, totaling points they award for the person's income level, credit history, current debt burden, and so on. The higher the point total, the more convinced the bank is that it's safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. We can think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the person's score falls below the minimum cutoff will the bank reject the null and deny the loan. When a person defaults on a loan, which type of error did the bank make? A) Type II error B) Type I error
A) Type II error
(d) A clean air standard requires that vehicle exhaust emissions not exceed specified limits for various pollutants. Many states require that cars be tested annually to be sure they meet these standards. Suppose state regulators double-check a random sample of cars that a suspect repair shop has certified as okay. They will revoke the shop's license if they find significant evidence that the shop is certifying vehicles that do not meet standards. Which type of error might environmentalists consider more serious? A) Type II error B) Type I error
A) Type II error
A human resource analyst wants to know if the applicants this year score, on average, higher on their placement exam than the 52.5 points the candidates averaged last year. She samples 50 recent tests and finds the average to be 54.1 points. She fails to reject the null hypothesis that the mean is 52.5 points. At the end of the year, they find that the candidates this year had a mean of 55.3 points. Has a Type I error or Type II error been made? Or no error? A) The analyst made a Type I error. The actual value was 55.3 points, which is close to 52.5. B) The analyst made a Type II error. The actual value was 55.3 points, which is greater than 52.5. C) The analyst did not make an error. The actual value was 55.3 points, which is close to 52.5. D) The analyst made a Type II error. The actual value was 55.3 points, which is close to 52.5. E) The analyst made a Type I error. The actual value was 55.3 points, which is greater than 52.5.
B) The analyst made a Type II error. The actual value was 55.3 points, which is greater than 52.5.
A bank wants to know if the enrollment on their website is above 30% based on a small sample of customers. They test Ho: p = 0.3 vs. Ha: p > 0.3 and reject the null hypothesis. Later they find out that actually 28% of all customers enrolled. Has a Type I error or Type II error been made? Or no error? A) The bank made a Type II error. The actual value is not greater than 0.3 but they rejected the null hypothesis. B) The bank made a Type I error. The actual value is not greater than 0.3 but they rejected the null hypothesis. C) The bank did not make an error. The actual value is not equal to 0.3 and they rejected the null hypothesis. D) The bank made a Type II error. The actual value is close to 0.3 but they rejected the null hypothesis. E) The bank made a Type I error. The actual value is close to 0.3 but they rejected the null hypothesis.
B) The bank made a Type I error. The actual value is not greater than 0.3 but they rejected the null hypothesis.
(c) Before lending someone money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the applicant, totaling points they award for the person's income level, credit history, current debt burden, and so on. The higher the point total, the more convinced the bank is that it's safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. We can think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the person's score falls below the minimum cutoff will the bank reject the null and deny the loan. Suppose the bank decides to lower the cutoff score from 250 points to 200. Is that analogous to choosing a higher or lower value of alpha for a hypothesis test? A) higher alpha level B) lower alpha level
B) lower alpha level
(d) Before lending someone money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the applicant, totaling points they award for the person's income level, credit history, current debt burden, and so on. The higher the point total, the more convinced the bank is that it's safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. We can think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the person's score falls below the minimum cutoff will the bank reject the null and deny the loan. What impact does this change in the cutoff value have on the chance of each type of error? A) Decreased Type I, decreased Type II. B) Increased Type I, increased Type II. C) Decreased Type I, increased Type II. D) Increased Type I, decreased Type II.
C) Decreased Type I, increased Type II.
(b) Most car engines need at least 87 octane to avoid "knocking" or "pinging," terms used to describe the pre-ignition that can happen when a fuel's octane is too low. An engineer is designing an experiment to raise the octane of an ethanol-based fuel. From previous studies, she thinks that with 8 experimental runs, she will have a power of 0.90 to detect a real increase of 3 points in the mean octane. If she wants the power to be the same, but she is interested in detecting an increase of only 1 point, what will she need to do? A) She will redesign the experiment, this time using a different type of fuel. B) She will reduce alpha while maximizing the power. C) She will use a larger sample size to keep the power the same. D) She will use a smaller sample size to keep the power the same.
C) She will use a larger sample size to keep the power the same.
A pharmaceutical company tests whether a drug lifts the headache relief rate from the 25% achieved by the placebo. They fail to reject the null hypothesis because the P-value is 0.465. Further testing shows that the drug actually relieves headaches in 38% of people. Has a Type I error or Type II error been made? Or no error? A) The company made a Type I error. It used a P-value to determine its conclusion instead of a critical value. B) The company made a Type I error. The null hypothesis was not rejected, but it was false. The true relief rate was greater than 0.25. C) The company made a Type II error. The null hypothesis was not rejected, but it was false. The true relief rate was greater than 0.25. D) The company did not make an error. They did not reject the null hypothesis when the P-value was almost 0.5. E) The company made a Type II error. It used a P-value to determine its conclusion instead of a critical value.
C) The company made a Type II error. The null hypothesis was not rejected, but it was false. The true relief rate was greater than 0.25.
(a) A clean air standard requires that vehicle exhaust emissions not exceed specified limits for various pollutants. Many states require that cars be tested annually to be sure they meet these standards. Suppose state regulators double-check a random sample of cars that a suspect repair shop has certified as okay. They will revoke the shop's license if they find significant evidence that the shop is certifying vehicles that do not meet standards. What is a Type I error? A) The regulators certify that the shop is meeting the standards when the shop is actually not meeting them. B) The regulators decide that the shop is not meeting the standards when the shop is not meeting them. C) The regulators decide that the shop is not meeting the standards when the shop is actually meeting them. D) The regulators certify that the shop is meeting the standards when the shop is meeting them.
C) The regulators decide that the shop is not meeting the standards when the shop is actually meeting them.
(a) Most car engines need at least 87 octane to avoid "knocking" or "pinging," terms used to describe the pre-ignition that can happen when a fuel's octane is too low. An engineer is designing an experiment to raise the octane of an ethanol-based fuel. From previous studies, she thinks that with 8 experimental runs, she will have a power of 0.90 to detect a real increase of 3 points in the mean octane. If the actual increase is only 1 point and all other things remain equal, will the power be increased or decreased? A) The power will remain constant, since the probability of incorrectly detecting the increase will remain the same. B) The power will be increased, since the probability of correctly detecting the increase will be greater. C) The power will be decreased, since the probability of incorrectly detecting the increase will be greater. D) The power will be decreased, since the probability of correctly detecting the increase will be lower. E) The power will be increased, since the probability of incorrectly detecting the increase will be lower. F) The power will remain constant, since the probability of correctly detecting the increase will remain the same.
D) The power will be decreased, since the probability of correctly detecting the increase will be lower.
(b) A clean air standard requires that vehicle exhaust emissions not exceed specified limits for various pollutants. Many states require that cars be tested annually to be sure they meet these standards. Suppose state regulators double-check a random sample of cars that a suspect repair shop has certified as okay. They will revoke the shop's license if they find significant evidence that the shop is certifying vehicles that do not meet standards. What is a Type II error? A) The regulators certify that the shop is meeting the standards when the shop is meeting them. B) The regulators decide that the shop is not meeting the standards when the shop is not meeting them. C) The regulators decide that the shop is not meeting the standards when the shop is actually meeting them. D) The regulators certify that the shop is meeting the standards when the shop is not actually meeting them.
D) The regulators certify that the shop is meeting the standards when the shop is not actually meeting them.
A student tests 100 students to determine whether other students on her campus prefer soda brand A or soda brand B and finds no evidence that preference for brand A is not 0.5. Later, a marketing company tests all students on campus and finds no difference. Has a Type I error or Type II error been made? Or no error? A) The student made a Type II error. She found no evidence that preference for brand A is not a certain value, but the marketing company found no evidence that preference for brand A is a certain value. B) The student made a Type II error. She did not reject the null hypothesis. C) The student made a Type I error. She found no evidence that preference for brand A is not a certain value, but the marketing company found no evidence that preference for brand A is a certain value. D) The student made a Type I error. She did not reject the null hypothesis. E) The student did not make an error. The actual value is 0.50, which was not rejected.
E) The student did not make an error. The actual value is 0.50, which was not rejected.