Can you please answer all the questions ?
HYPOTHESIS TESTING Dr. V.R. Bencivenga Economic Statistics Economics 329 CANVAS QUIZ #1 6 Hypothesis testing, one population (Topic 9) HYPOTHESIS TESTING INSTRUCTIONS In the questions of this quiz, here are the answer choices for when you are asked about the population and the sampling distribution. When you are asked “What is the population?”, give your answer choice (letter from A to C) from the following: A . Bernoulli. Xi, i = 1,…,n are i.i.d. Bernoulli random variables with success probability p B. Normal. Xi, i = 1,…,n are i.i.d. N(  ,  2) C. General (arbitrary). X i, i = 1,…,n are i.i.d. with E[X i] =  and Var(X i) =  2 Instructions continue. HYPOTHESIS TESTING When you are asked “ What is the sampling distribution (and why)? ”, give your answe r choice (letter from A to G ) from among the following: A. . Random sample from a normal population (known variance) B. . Random sample from a normal population (unknown variance) C. . Random sample from a normal population D. . Central Limit Theorem (large random sample from a population that is not normal, and which may be a general or arbitrary or unknown population) E. X ~ binomial(n, p). Sm all random sample from a Bernoulli population F. . Central Limit Theorem (large random sample from a Bernoulli population) G . . Central Limit Theorem (large random sample from a Bernoulli population)   X ~ N , n   01 n1 X ~t s n     2 2 n1 2 n 1 s ~       XX or ~ approximately N 0,1 s nn          pp ~ approximately N 0,1 p 1 p n       pp ~ approximately N 0,1 p 1 p n   HYPOTHESIS TESTING MATH SAT SCORES 1. This is the first of 13 questions in this set. NOTE: This question set follows up on a question set in Quiz #14. The College Board wants to calibrate the difficulty of the math SAT test. Difficulty is interprete d as the population mean score. The College Board’s statisticians administer the test to 41 randomly -selected students. The mean score in the sample is 534. The statisticians are willing to a ssume that scores on the math SAT are approximately normally d istributed, and that the population sta ndard deviation is 100 points. The SAT authors are aiming for a population mean score of 525. They use their sample to test the null hypothesis that the population mean is 525, against the alternative hypothesis th at it is not 525, using a 1% level of significance. Which sampling distribution is the basis for this hypothesis test? Letter (see multiple choices in the instructions) HYPOTHESIS TESTING 2. Begin by using the rejection region approach. What is the f orm of the rejection region? Choose the letter that represents your choice. A . Reject H 0 if H 0 if or if B. Reject H 0 if C. Reject H 0 if 3. Give the value of the critical value (cv) if the rejection region has one critical value or the value of the smaller critical value (cv 1) if the rejection region has two critical values. One decimal 4. Give the value of the larger critical value (cv 2) if the r ejection region has two critical values. If the rejection region has only one critical value, enter NA. One decimal 5. What is the conclusion of the test? a. Fail to reject H 0 (accept H 0) b. Reject H 0 in favor of H A 1 X cv  2 X cv  X cv  X cv  HYPOTHESIS TESTING 6. What is the probability of Type II error if the population mean is 5 10? Four decimals 7. Continued. What is the power of the test if the population mean is 5 10? Four decimals 8. What is the probability of Type II error if the population mean is 5 40? Four decim als 9. Continued. What is the power of the test if the population mean is 5 40? Four decimals HYPOTHESIS TESTING 10. What is the shape of th e rejection probability curve? Choose the letter that represents your choice. A. B. C. μ P(H 0 is rejected | μ) μ μ P(H 0 is rejected | μ) P(H 0 is rejected | μ) HYPOTHESIS TESTING 11 . Now let’s use the p -value approach. Calculate the p -value. Four decimals 12 . Which of the following describes the criterion for rejecting the null hypothesis using the p – value approach? a. Reject H 0 if th e p -value is larger than the significance level. b. Reject H 0 if the p -value is smaller than the significance level. 13 . Now let’s use the confidence interval approach. Consider the confidence interval you calculated for this problem set in Quiz #14a. Which of the following describes the criterion for rejecting the null hypothesis using the confidence interval approach? a. Reject H 0 if the confidence interval contains 525. b. Accept H 0 if the confidence interval contains 525. HYPOTHESIS TESTING DRU G TRIAL 14. This is the first of 9 questions in this set. A pharmaceutical company plans to conduct a small trial of an experimental drug. The drug will be administered to 15 patients. For each patient, it will be recorded whether the drug cured the dis ease or not. The pharmaceutical company’s statistician wants to test the null hypothesis that the cure rate of this experimental drug is no greater than 0.8 against the alternative hypothesis that the cure rate is greater than 0.8. The null and alternati ve are chosen to put the burden on the data to convince the statistician that the drug’s cure rate is high. The statistician decides to use the following rejection region: Reject H 0 if X > 12, where X is the number of patients who are cured. What is the probability H 0 will be rejected if p = 0.8? Four decimals 15 . The probability you calculated in the previous question is a. probability of Type I error b. power c. probability of Type II error HYPOTHESIS TESTING 16 . What is the probability H 0 will be rejected if p = 0.75? Four decimals 17 . The probability you calculated in the previous question is a. probability of Type I error b. power c. probability of Type II error 18 . What is the probability H 0 will be rejected if p = 0.85? Four decimals 19 . The probability you calculated in the previous question is a. probability of Type I error b. power c. probability of Type II error HYPOTHESIS TESTING 20 . What is the probability H 0 will be rejected if p = 0.9? Four decimals 21 . The probability you calcul ated in the previous question is a. probability of Type I error b. power c. probability of Type II error HYPOTHESIS TESTING 22 . Make a rough sketch of the rejection probability curve. Sketch the four points on the rejection probability curve that you calculated in the questions above. Put p on the horizontal axis. Put the probability that H 0 will be rejected on the vertical axis. Indicate the interval of values of p in H 0, and the interval of values of p in H A. Indicate where on the graph we find t he following probabilities: probability of Type I error, power, probability of Type II error, probability H 0 will be correctly accepted. Based on your sketch (which will help you understand how this hypothesis test works), which of the following statemen ts is (are) TRUE ? Multiple answers – Check all of the correct answers to get credit for this question. The probability of Type I error is largest when p = 0.8. Power increases as p increases above 0.8. The probability of Type II error increases as p appro aches 0.8 from above (from the right). The probability that H 0 will be correctly accepted increases as p decreases below 0.8. Probability of Type I error + probability of Type II error = 1 Probability of Type I error decreases as p increases above 0.8. The probability H 0 will be rejected decreases as p increases. Power decreases as p decreases below 0.8. HYPOTHESIS TESTING JET ENGINES 23 . This is the first of 9 questions in this set. A manufacturer of passenger airplanes has, in the past, put jet engine s on its planes that had a TBO (“time between overhaul”) of 15,000 flight hours. Twenty percent of these engines required an overhaul before 15,000 flight hours. The airplane manufacturer is considering a different jet engine for its planes, in part beca use they believe the probability an engine of this different type will require an overhaul before 15,000 flight hours is less than 0.2. The airplane manufacturer wants to gather evidence on this issue. They want to design a test of the null hypothesis th at the probability an engine (of the different type they are considering) will require an early overhaul is at least 0.2, against the alternative hypothesis that it is less. (The airplane manufacturer quite correctly wants to put the burden on the data to convince them that the probability of an early overhaul is less than 0.2.) Suppose a random sample of 10 engines will be tested. Each engine will be used for 15,000 flight hours and it will be recorded whether the engine required an overhaul before 15,0 00 flight hours or not. What is the population? Letter (see multiple choices in the instructions) HYPOTHESIS TESTING 24. Which sampling distribution is the basis for this hypothesis test? Letter (see multiple choices in the instructions) 25 . Which o f the following gives the form of the null and alternative hypothesis? Choose the letter that represents your choice. A . H 0: p = p 0 vs. H A: p  p0 B. H 0: p  p0 vs H A: p > p 0 C. H 0: p  p0 vs H A: p < p 0 26 . What is t he form of the rejection region? ( cv is the critical value.) a. Reject H 0 if X < cv b. Reject H 0 if X > cv c. Reject H 0 if X < cv 1 or if X > cv 2 HYPOTHESIS TESTING 27 . Figure out what the smallest rejection region is. What is the critical value (cv) for the smallest rejection region? HINT: Be careful! Remember that, by convention, the critical value is included in the acceptance region . Therefore, the rejection region is described using a strict inequality. Integer 28 . What is the smallest value of the maximum probability of Type I error? Four decimals HINT: Divide this question into two steps. First, figure out the value of p for which the probability of Type I error is largest. Then (using that value of p) calculate the probability H 0 will be rejected using the smallest rej ection region. 29 . The airplane manufacturer does not like the answer to the previous question. They think that the smallest value of the maximum probability of Type I error is too large. They don’t want it to exceed 0.05. What is the minimum sample si ze the airplane manufacturer must use? Integer HYPOTHESIS TESTING 30 . Now let’s consider a different sample size. Now suppose a random sample of 2 5 engines will be tested (instead of 10). What is the smallest value of the maximum probability of Type I error? Four decimals HINT: This is the same question as you answered above, but with a larger sample size. HYPOTHESIS TESTING 31 . The airplane manufacturer is willing to accept a maximum probability of Type I error of 0.05. Compared to your answe r in the previous question, can you make the rejection region larger (a larger set of values)? Find the largest rejection region where the maximum probability of Type I error is less than or equal to 0.05. Specifically, what is the value of cv in that re jection region? What is the value of cv in the largest rejection region where the maximum probability of Type I error is less than or equal to 0.05? Integer HINT #1: To repeat the hint from above, remember that the critical value is included in the ac ceptance region. Therefore, the rejection region is described using a strict inequality. HINT #2: The purpose of this question is for you to understand the following idea. At a given sample size, there is a tradeoff between the probability of Type I er ror (probability of incorrectly rejecting H 0) and power (probability of correctly rejecting H 0). We can reduce the maximum probability of Type I error by making the rejection region smaller (a smaller set of values of the test statistic). But if we do th at, power decreases, too. That’s why, in general, we want to use the largest rejection region that does not push the maximum probability of Type I error over the level we’ve decided is tolerable. So what you need to do to answer this question is expand t he rejection region (make it a larger set of values), until the maximum probability of Type I error goes over 0.05. Then “back up” to the largest rejection region where the maximum probability of Type I error is less than or equal to 0.05. HYPOTHESIS TES TING BELIEF ABOUT THE IMPORTANCE OF WEARING A FACE MASK IN PUBLIC TO SLOW THE SPREAD OF THE CORONAVIRUS 32 . This is the first of 5 questions in this set. A researcher wants to test the null hypothesis that the population proportion of people who believe wearing a face mask in public is an important public health measure is at least 0.6, against the alternative hypothesis that it is less. A 5% level of significance will be used. The researcher plans to poll a random sample of 2,000 adults. What is the population? Letter (see multiple choices in the instructions) 33 . The researcher wants to use an approximate sampling distribution based on the Central Limit Theorem. Which sampling distribution should the researcher use ? Letter (see multiple choices i n the instructions) HYPOTHESIS TESTING 34 . The rejection region is of the form: Reject H 0 if the sample proportion is less than cv. Calculate cv. Four decimals HINT: Use the “boundary value” of p when you calculate cv. In the previous two questio n sets, the probability of Type I error is at its largest when p is on the boundary between H0 and HA. That is not necessarily true when n is large and we use an approximate sampling distribution based on the Central Limit Theorem. Never theless, it is co nventional to use the boundary value of p to calculate cv, and that is what we will do. Implicitly (and in some textbooks), the null hypothesis becomes the simple hypothesis H 0: p = boundary value. 35 . In the sample, 1180 of 2,000 say wearing a face mask is important. Calculate the p -value. Four decimals 36 . What is the conclusion of the test? a. Fail to reject H 0 (accept H 0) b. Reject H 0 in favor of H A HYPOTHESIS TESTING HOUSING MARKET 37 . This is the first of 8 questions in this set. NOTE: Thi s question set follows up on a question set in Quiz #14. In a certain city, the number of days a house is on the market before it is sold is approximately normally distributed. In a random sample of 21 houses, the mean number of days before the sale was 94, and the standard deviation was 27 days. A realty company wants to test the null hypothesis that the population mean is 100 days, against the alternative hypothesis that it is not, using a 10% significance level. What is the value of cv 1, the lower cr itical value? Two decimals 38 . What is the value of cv 2, the upper critical value? Two decimals 39 . What is the conclusion of the test? a. Fail to reject H 0 (accept H 0) b. Reject H 0 in favor of H A HYPOTHESIS TESTING 40 . The realty company also wants to us e this sample to test the null hypothesis that the standard deviation of the number of days on the market equals 2 1 (that the variance is 4 41 squared days) , against the alternative hypothesis that it does not, usin g a 10% level of significance . Which sampling distribution is the basis for this hypoth esis test? Letter (see multiple choices in the instructions) 41 . What is the form of the rejection region? Choose the letter that represents your choice. A . Reject H 0 if s 2 < cv 1 or if s 2 > cv 2 B. Reject H 0 if s 2 < cv C. Reject H 0 if s 2 > cv HYPOTHESI S TESTING 42 . Give the value of the critical value (cv) if the rejection region has one critical value or the value of the smaller critical value (cv 1) if the rejection region has two critical values. One decimal 43 . Give the value of the larger critic al value (cv 2) if the rejection region has two critical values. If the rejection region has only one critical value, enter NA. One decimal 44 . What is the conclusion of the test? a. Fail to reject H 0 (accept H 0) b. Reject H 0 in favor of H A HYPOTHESIS TESTING AGE OF INSURANCE POLICY HOLDERS 45 . This is the first of 5 questions in this set. NOTE: This question set follows up on a question set in Quiz #14. A large automobile insurance company wants to test the null hypothesis that the mean age in its population of policyholders is 50, against the alternative hypothesis that it is different from 50. In a random sample of 361 policyholders, the average age is 47.2 years, and the variance is 121 (squared years). The significance level is 5%. What is the population? Letter (see multiple choices in the instructions) 46 . Which sampling distribution is the basis for this hypothesis test? Letter (see multiple choices in the instructions) 47 . Give the value of the critical value (cv) if the rejection regi on has one critical value or the value of the smaller critical value (cv 1) if the rejection region has two critical values. One decimal 48 . Give the value of the larger critical value (cv 2) if the rejection region has two critical values. If the rejectio n region has only one critical value, enter NA. One decimal 49 . What is the conclusion of the test? a. Fail to reject H 0 (accept H 0) b. Reject H 0 in favor of H A