10/29/2015 Week 6 Assignment YINWEI LIU CED 6030, section 02, Fall 2015 Instructor: He Wang WebAssign Week 6 Assignment (Homework) Current Score : - / 114 Due : Wednesday, November 4 2015 11:59 PM EST 0/2 submissions 1. -/6 pointsBBUnderStat11 8.1.001. Briefly answer the following questions. (a) What is a null hypothesis H0? A specific hypothesis where the claim is that the population parameter does not equal 0. A specific hypothesis where the claim is that the population parameter is equal to 0. A working hypothesis making a claim about the population parameter in question. Any hypothesis that differs from the original claim being made. (b) What is an alternate hypothesis H1? A working hypothesis making a claim about the population parameter in question. Any hypothesis that differs from the original claim being made. A specific hypothesis where the claim is that the population parameter does not equal 0. A specific hypothesis where the claim is that the population parameter is equal to 0. (c) What is a type I error? Type I error is rejecting the null hypothesis when it is false. Type I error is failing to reject the null hypothesis when it is false. Type I error is rejecting the null hypothesis when it is true. Type I error is failing to reject the null hypothesis when it is true. What is a type II error? Type II error is rejecting the null hypothesis when it is false. Type II error is failing to reject the null hypothesis when it is false. Type II error is rejecting the null hypothesis when it is true. Type II error is failing to reject the null hypothesis when it is true. (d) What is the level of significance of a test? The probability of a type I error. The probability of a type II error. What is the probability of a type II error? 1 1 https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 1/26 10/29/2015 Week 6 Assignment 2. -/1 pointsBBUnderStat11 8.1.002. In a statistical test, we have a choice of a lefttailed test, a righttailed test, or a twotailed test. Is it the null hypothesis or the alternate hypothesis that determines which type of test is used? Explain your answer. The alternative hypothesis because it specifies what the level of significance of the test will be. The alternative hypothesis because it specifies the region of interest for the parameter in question. The null hypothesis because it specifies what the level of significance of the test will be. The null hypothesis because it specifies the region of interest for the parameter in question. 3. -/1 pointsBBUnderStat11 8.1.003. If we fail to reject (i.e., "accept") the null hypothesis, does this mean that we have proved it to be true beyond all doubt? Explain your answer. Yes, if we fail to reject the null we have found evidence that the null is true beyond all doubt. No, it suggests that the evidence is not sufficient to merit rejecting the null hypothesis. No, it suggests that the null hypothesis is true only some of the time. Yes, it suggests that the evidence is sufficient to merit rejecting the alternative hypothesis beyond all doubt. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 2/26 10/29/2015 Week 6 Assignment 4. -/5 pointsBBUnderStat11 8.1.015. The body weight of a healthy 3 monthold colt should be about = 60 kg. (Source: The Merck Veterinary Manual, a standard reference manual used in most veterinary colleges.) (a) If you want to set up a statistical test to challenge the claim that = 60 kg, what would you use for the null hypothesis H0? < 60 = 60 > 60 60 (b) In Nevada, there are many herds of wild horses. Suppose you want to test the claim that the average weight of a wild Nevada colt (3 months old) is less than 60 kg. What would you use for the alternate hypothesis H1? < 60 = 60 60 > 60 (c) Suppose you want to test the claim that the average weight of such a wild colt is greater than 60 kg. What would you use for the alternate hypothesis? 60 = 60 > 60 < 60 (d) Suppose you want to test the claim that the average weight of such a wild colt is different from 60 kg. What would you use for the alternate hypothesis? < 60 = 60 > 60 60 (e) For each of the tests in parts (b), (c), and (d), would the area corresponding to the Pvalue be on the left, on the right, or on both sides of the mean? both left right left right both right left both left both right https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 3/26 10/29/2015 Week 6 Assignment 5. -/8 pointsBBUnderStat11 8.1.016. How much customers buy is a direct result of how much time they spend in the store. A study of average shopping times in a large national houseware store gave the following information (Source: Why We Buy: The Science of Shopping by P. Underhill). Women with female companion: 8.3 min. Women with male companion: 4.5 min. Suppose you want to set up a statistical test to challenge the claim that a woman with a female friend spends an average of 8.3 minutes shopping in such a store. (a) What would you use for the null and alternate hypotheses if you believe the average shopping time is less than 8.3 minutes? Ho: < 8.3 H1: = 8.3 Ho: = 8.3 H1: < 8.3 Ho: = 8.3 H1: 8.3 Ho: = 8.3 H1: > 8.3 Is this a righttailed, lefttailed, or twotailed test? twotailed lefttailed righttailed (b) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from 8.3 minutes? Ho: 8.3 H1: = 8.3 Ho: = 8.3 H1: < 8.3 Ho: = 8.3 H1: > 8.3 Ho: = 8.3 H1: 8.3 Is this a righttailed, lefttailed, or twotailed test? twotailed lefttailed righttailed Stores that sell mainly to women should figure out a way to engage the interest of men! Perhaps comfortable seats and a big TV with sports programs. Suppose such an entertainment center was installed and you now wish to challenge the claim that a woman with a male friend spends only 4.5 minutes shopping in a houseware store. (c) What would you use for the null and alternate hypotheses if you believe the average shopping time is more than 4.5 minutes? Ho: = 4.5 H1: 4.5 Ho: > 4.5 H1: = 4.5 Ho: = 4.5 H1: > 4.5 Ho: = 4.5 H1: < 4.5 Is this a righttailed, lefttailed, or twotailed test? lefttailed twotailed righttailed https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 4/26 10/29/2015 Week 6 Assignment (d) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from 4.5 minutes? Ho: = 4.5 H1: > 4.5 Ho: 4.5 H1: = 4.5 Ho: = 4.5 H1: < 4.5 Ho: = 4.5 H1: 4.5 Is this a righttailed, lefttailed, or twotailed test? twotailed lefttailed righttailed https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 5/26 10/29/2015 Week 6 Assignment 6. -/8 pointsBBUnderStat11 8.1.020. Gentle Ben is a Morgan horse at a Colorado dude ranch. Over the past 8 weeks, a veterinarian took the following glucose readings from this horse (in mg/100 ml). 93 88 82 103 101 112 82 87 The sample mean is x 93.5. Let x be a random variable representing glucose readings taken from Gentle Ben. We may assume that x has a normal distribution, and we know from past experience that = 12.5. The mean glucose level for horses should be = 85 mg/100 ml. Do these data indicate that Gentle Ben has an overall average glucose level higher than 85? Use = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a lefttailed, righttailed, or twotailed test? H0: = 85 H1: 85 twotailed H0: = 85 H1: < 85 lefttailed H0: = 85 H1: > 85 righttailed H0: > 85 H1: = 85 righttailed (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since we assume that x has a normal distribution with unknown . The Student's t, since n is large with unknown . The Student's t, since we assume that x has a normal distribution with known . The standard normal, since we assume that x has a normal distribution with known . What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find (or estimate) the Pvalue. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the Pvalue. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 6/26 10/29/2015 Week 6 Assignment (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? At the = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) State your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that Gentle Ben's glucose is higher than 85 mg/100 ml. There is insufficient evidence at the 0.05 level to conclude that Gentle Ben's glucose is higher than 85 mg/100 ml. 7. -/1 pointsBBUnderStat11 8.2.001. For the same sample data and null hypothesis, how does the Pvalue for a twotailed test of compare to that for a onetailed test? The Pvalue for a twotailed test is half the Pvalue for a onetailed test. The Pvalue for a twotailed test is three times the Pvalue for a onetailed test. The Pvalue for a twotailed test is the same as the Pvalue for a onetailed test. The Pvalue for a twotailed test is twice the Pvalue for a onetailed test. 8. -/1 pointsBBUnderStat11 8.2.002. To test for an x distribution that is moundshaped using sample size n 30, how do you decide whether to use the normal or Student's t distribution? If is unknown, use the standard normal distribution. If is known, use the Student's t distribution with n - 1 degrees of freedom. If is known, use the standard normal distribution. If is unknown, use the Student's t distribution with n - 1 degrees of freedom. If is known, use the standard normal distribution. If is unknown, use the Student's t distribution with n degrees of freedom. For large samples we always the standard normal distribution. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 7/26 10/29/2015 Week 6 Assignment 9. -/1 pointsBBUnderStat11 8.2.003. When using the Student's t distribution to test , what value do you use for the degrees of freedom? n n + 1 n - 2 n - 1 10.-/1 pointsBBUnderStat11 8.2.004. Consider a test for . If the Pvalue is such that you can reject H0 at the 5% level of significance, can you always reject H0 at the 1% level of significance? Explain your answer. No. If the Pvalue lies between 0.01 and 0.05 you would reject at the 5% level of significance, but not at the 1% level. No. If the Pvalue lies above 0.01 you would reject at the 5% level of significance, but not at the 1% level. Yes. If H0 is rejected at the 5% level it will always be rejected at the 1% level. No. If the Pvalue lies above 0.05 you would reject at the 5% level of significance, but not at the 1% level. 11.-/1 pointsBBUnderStat11 8.2.006. If sample data is such that for a onetailed test of you can reject H0 at the 1% level of significance, can you always reject H0 for a twotailed test at the same level of significance? Explain your answer. Yes. If H0 for a onetailed test is rejected at the 1% level of significance, it will always be rejected for a twotailed test at the same level of significance. No. If H0 for a onetailed test is rejected at the 1% level of significance, it will never be rejected for a twotailed test at the same level of significance. No. The Pvalue for the twotailed test could be greater than 0.01. Yes. The Pvalue for the twotailed test could be greater than 0.01. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 8/26 10/29/2015 Week 6 Assignment 12.-/8 pointsBBUnderStat11 8.2.013. A random sample of 46 adult coyotes in a region of northern Minnesota showed the average age to be x = 2.03 years, with sample standard deviation s = 0.78 years. However, it is thought that the overall population mean age of coyotes is = 1.75. Do the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: = 1.75 yr H1: 1.75 yr H0: = 1.75 yr H1: < 1.75 yr H0: < 1.75 yr H1: = 1.75 yr H0: = 1.75 yr H1: > 1.75 yr H0: > 1.75 yr H1: = 1.75 yr (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The Student's t, since the sample size is large and is known. The standard normal, since the sample size is large and is known. The Student's t, since the sample size is large and is unknown. The standard normal, since the sample size is large and is unknown. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Find the Pvalue. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the Pvalue. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 9/26 10/29/2015 Week 6 Assignment (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? At the = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that coyotes in the specified region tend to live longer than 1.75 years. There is insufficient evidence at the 0.01 level to conclude that coyotes in the specified region tend to live longer than 1.75 years. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 10/26 10/29/2015 Week 6 Assignment 13.-/10 pointsBBUnderStat11 8.2.018. Let x be a random variable that represents hemoglobin count (HC) in grams per 100 milliliters of whole blood. Then x has a distribution that is approximately normal, with population mean of about 14 for healthy adult women. Suppose that a female patient has taken 10 laboratory blood tests during the past year. The HC data sent to the patient's doctor are as follows. 14 19 15 18 13 12 13 17 16 10 (i) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.) x= s= (ii) Does this information indicate that the population average HC for this patient is higher than 14? Use = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: > 14 H1: = 14 H0: = 14 H1: < 14 H0: = 14 H1: > 14 H0: < 14 H1: = 14 H0: = 14 H1: 14 (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since we assume that x has a normal distribution and is unknown. The Student's t, since we assume that x has a normal distribution and is known. The Student's t, since we assume that x has a normal distribution and is unknown. The standard normal, since we assume that x has a normal distribution and is known. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Find the Pvalue. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the Pvalue. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 11/26 10/29/2015 Week 6 Assignment (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? At the = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the population average HC for this patient is higher than 14. There is insufficient evidence at the 0.01 level to conclude that the population average HC for this patient is higher than 14. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 12/26 10/29/2015 Week 6 Assignment 14.-/9 pointsBBUnderStat11 8.2.026. Is there a relationship between confidence intervals and twotailed hypothesis tests? Let c be the level of confidence used to construct a confidence interval from sample data. Let be the level of significance for a twotailed hypothesis test. The following statement applies to hypothesis tests of the mean. For a twotailed hypothesis test with level of significance and null hypothesis H0: = k, we reject H0 whenever k falls outside the c = 1 - confidence interval for based on the sample data. When k falls within the c = 1 - confidence interval, we do not reject H0. (A corresponding relationship between confidence intervals and twotailed hypothesis tests also is valid for other parameters, such as p, 1 2, or p1 p2, which we will study in later sections.) Whenever the value of k given in the null hypothesis falls outside the c = 1 - confidence interval for the parameter, we reject H0. For example, consider a twotailed hypothesis test with = 0.05 and H0: = 21 H1: 21 A random sample of size 23 has a sample mean x = 19 from a population with standard deviation = 5. (a) What is the value of c = 1 ? Using the methods of Chapter 7, construct a 1 confidence interval for from the sample data. (Round your answers to two decimal places.) lower limit upper limit What is the value of given in the null hypothesis (i.e., what is k)? k = Is this value in the confidence interval? Yes No Do we reject or fail to reject H0 based on this information? Fail to reject, since = 21 is not contained in this interval. Fail to reject, since = 21 is contained in this interval. Reject, since = 21 is not contained in this interval. Reject, since = 21 is contained in this interval. (b) Using methods of Chapter 8, find the Pvalue for the hypothesis test. (Round your answer to four decimal places.) Do we reject or fail to reject H0? Reject the null hypothesis, there is sufficient evidence that differs from 21. Fail to reject the null hypothesis, there is insufficient evidence that differs from 21. Fail to reject the null hypothesis, there is sufficient evidence that differs from 21. Reject the null hypothesis, there is insufficient evidence that differs from 21. Compare your result to that of part (a). We rejected the null hypothesis in part (a) but failed to reject the null hypothesis in part (b). These results are the same. We rejected the null hypothesis in part (b) but failed to reject the null hypothesis in part (a). https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 13/26 10/29/2015 Week 6 Assignment 15.-/4 pointsBBUnderStat11 8.2.029. The Student's t distribution table gives critical values for the Student's t distribution. Use an appropriate d.f. as the row header. For a righttailed test, the column header is the value of found in the onetail area row. For a lefttailed test, the column header is the value of found in the onetail area row, but you must change the sign of the critical value t to t. For a twotailed test, the column header is the value of from the twotail area row. The critical values are the t values shown. A random sample of 51 adult coyotes in a region of northern Minnesota showed the average age to be x = 2.07 years, with sample standard deviation s = 0.81 years. However, it is thought that the overall population mean age of coyotes is = 1.75. Do the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use = 0.01. Solve the problem using the critical region method of testing (i.e., traditional method). (Round your answers to three decimal places.) test statistic = critical value = State your conclusion in the context of the application. Fail to reject the null hypothesis, there is sufficient evidence that the average age of Minnesota coyotes is higher than 1.75 years. Reject the null hypothesis, there is insufficient evidence that the average age of Minnesota coyotes is higher than 1.75 years. Fail to reject the null hypothesis, there is insufficient evidence that the average age of Minnesota coyotes is higher than 1.75 years. Reject the null hypothesis, there is sufficient evidence that the average age of Minnesota coyotes is higher than 1.75 years. Compare your conclusion with the conclusion obtained by using the Pvalue method. Are they the same? We reject the null hypothesis using the traditional method, but fail to reject using the Pvalue method. We reject the null hypothesis using the Pvalue method, but fail to reject using the traditional method. The conclusions obtained by using both methods are the same. 16.-/1 pointsBBUnderStat11 8.3.001. To use the normal distribution to test a proportion p, the conditions np > 5 and nq > 5 must be satisfied. Does the value of p come from H0, or is it estimated by using p from the sample? The value of p is estimated using p from the sample. The value of p comes from H0. Neither, the value of p is guessed by the person in charge of the study. The value of p comes from both H0 and p . https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 14/26 10/29/2015 Week 6 Assignment 17.-/1 pointsBBUnderStat11 8.3.002. Consider a binomial experiment with n trials and r successes. To construct a test for a proportion p, what value do we use for the sample test statistic? n / r r / n (r / n)2 2r / n 18.-/1 pointsBBUnderStat11 8.3.003. In general, if sample data are such that the null hypothesis is rejected at the = 1% level of significance based on a twotailed test, is H0 also rejected at the = 1% level of significance for a corresponding onetailed test? Explain your answer. Yes. If the twotailed Pvalue is smaller than , the onetailed area will be larger than . Yes. If the twotailed Pvalue is smaller than , the onetailed area is also smaller than . No. If the twotailed Pvalue is smaller than , the onetailed area will be larger than . No. If the twotailed Pvalue is smaller than , the onetailed area is also smaller than . 19.-/5 pointsBBUnderStat11 8.3.022. Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 32 arrests last month, 23 were of males aged 15 to 34 years. Use a 1% level of significance to test the claim that the population proportion of such arrests is the city different from 70%. Solve the problem using both the traditional method and the Pvalue method. Since the sampling distribution of p is the normal distribution, you can use critical values from the standard normal distribution as shown in the table of critical values of the z distribution. (Round the test statistic and the critical value to two decimal places. Round the Pvalue to four decimal places.) test statistic = critical value = Pvalue = State your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the true proportion of such arrests in the city differs from 70%. There is insufficient evidence at the 0.01 level to conclude that the true proportion of such arrests in the city differs from 70%. Compare your conclusion with the conclusion obtained by using the Pvalue method. Are they the same? We reject the null hypothesis using the traditional method, but fail to reject using the Pvalue method. The conclusions obtained by using both methods are the same. We reject the null hypothesis using the Pvalue method, but fail to reject using the traditional method. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 15/26 10/29/2015 Week 6 Assignment 20.-/1 pointsBBUnderStat11 8.4.001. Are data that can be paired independent or dependent? ---Select--- 21.-/1 pointsBBUnderStat11 8.4.002. Consider a set of data pairs. What is the first step in processing the data for a paired differences test? What is the symbol for the sample test statistic? Describe the value of the sample test statistic. Take the squared differences. The test statistic symbol is d, which is the mean of the squared differences. Take the paired differences. The test statistic symbol is x, which is the mean of the paired differences. Take the paired differences. The test statistic symbol is d, which is the sample standard deviation of the paired differences. Take the paired differences. The test statistic symbol is d, which is the mean of the paired differences. 22.-/1 pointsBBUnderStat11 8.4.003. When testing the difference of means for paired data, what is the null hypothesis? Ho: d < 0 Ho: d > 0 Ho: d = 0 Ho: d 0 23.-/1 pointsBBUnderStat11 8.4.004. When conducting a paired differences test, what is the value of n? The total number of data values. The total number of data values minus 1. The number of data pairs. The number of data pairs minus 1. 24.-/1 pointsBBUnderStat11 8.4.005. When using a Student's t distribution for a paired differences test with n data pairs, what value do you use for the degrees of freedom? n - 1 n 2n 2n - 1 https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 16/26 10/29/2015 Week 6 Assignment 25.-/2 pointsBBUnderStat11 8.4.006. Alisha is conducting a paired differences test for a "before (B score) and after (A score)" situation. She is interested in testing whether the average of the "before" scores is higher than that of the "after" scores. (a) To use a righttailed test, how should Alisha construct the differences between the "before" and "after" scores? d = B + A d = A + B d = A - B d = B - A (b) To use a lefttailed test, how should she construct the differences between the "before" and "after" scores? d = A - B d = B - A d = A + B d = B + A https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 17/26 10/29/2015 Week 6 Assignment 26.-/8 pointsBBUnderStat11 8.4.009. In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the Pvalue by a small amount and therefore produce a slightly more "conservative" answer. Are America's top chief executive officers (CEOs) really worth all that money? One way to answer this question is to look at row B, the annual company percentage increase in revenue, versus row A, the CEO's annual percentage salary increase in that same company. Suppose a random sample of companies yielded the following data: B: Percent increase for company A: Percent increase for CEO 16 20 26 18 6 4 21 37 24 30 29 14 4 19 15 30 Do these data indicate that the population mean percentage increase in corporate revenue (row B) is different from the population mean percentage increase in CEO salary? Use a 5% level of significance. (Let d = B A.) (a) What is the level of significance? State the null and alternate hypotheses. H0: d = 0 H1: d < 0 H0: d 0 H1: d = 0 H0: d > 0 H1: d = 0 H0: d = 0 H1: d > 0 H0: d = 0 H1: d 0 (b) What sampling distribution will you use? What assumptions are you making? The Student's t. We assume that d has an approximately uniform distribution. The standard normal. We assume that d has an approximately normal distribution. The Student's t. We assume that d has an approximately normal distribution. The standard normal. We assume that d has an approximately uniform distribution. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Find the Pvalue. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the Pvalue. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 18/26 10/29/2015 Week 6 Assignment (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? Since the Pvalue > , we reject H0. The data are not statistically significant. Since the Pvalue > , we fail to reject H0. The data are not statistically significant. Since the Pvalue , we fail to reject H0. The data are statistically significant. Since the Pvalue , we reject H0. The data are statistically significant. (e) Interpret your conclusion in the context of the application. Reject H0. At the 5% level of significance, the evidence is insufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary. Fail to reject H0. At the 5% level of significance, the evidence is sufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary. Reject H0. At the 5% level of significance, the evidence is sufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary. Fail to reject H0. At the 5% level of significance, the evidence is insufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary. 27.-/1 pointsBBUnderStat11 8.5.002. Consider a hypothesis test of difference of means for two independent populations x1 and x2. Suppose that both sample sizes are greater than 30 and that you know 1 but not 2. Is it standard practice to use the normal distribution or a Student's t distribution? Use the Student's t distribution because we do not know 2. Use the normal distribution because both sample sizes are greater than 30. Use the Student's t distribution because both sample sizes are greater than 30. Use the normal distribution because we do not know 2. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 19/26 10/29/2015 Week 6 Assignment 28.-/1 pointsBBUnderStat11 8.5.003. Consider a hypothesis test of difference of means for two independent populations x1 and x2. What are two ways of expressing the null hypothesis? H0: 1 = 2 or H0: 1 + 2 = 0 H0: 1 < 2 or H0: 1 - 2 < 0 H0: 1 = 2 or H0: 1 - 2 = 0 H0: 1 > 2 or H0: 1 - 2 > 0 29.-/1 pointsBBUnderStat11 8.5.005. Consider a hypothesis test of difference of proportions for two independent populations. Suppose random samples produce r1 successes out of n1 trials for the first population and r2 successes out of n2 trials for the second population. What is the best pooled estimate p for the population probability of success using H0: p1 = p2? (r1 r2) / (n1 + n2) (r1 r2) / (n1 n2) (r1 + r2) / (n1 n2) (r1 + r2) / (n1 + n2) 30.-/2 pointsBBUnderStat11 8.5.006. Consider use of a Student's t distribution to test the difference of means for independent populations using random samples of sizes n1 and n2 . (a) Which process gives the larger degrees of freedom, Satterthwaite's approximation or using the smaller of n1 - 1 and n2 - 1? Which method is more conservative? What do we mean by "conservative"? Note that computer programs and other technology commonly use Satterthwaite's approximation. Satterthwaite's approximation is larger. This method is more conservative because it uses a larger degrees of freedom. Satterthwaite's approximation is smaller. Using the smaller of n1 - 1 and n2 - 1 is more conservative because it uses a larger degrees of freedom. Satterthwaite's approximation is smaller. This method is more conservative because it uses a smaller degrees of freedom. Satterthwaite's approximation is larger. Using the smaller of n1 - 1 and n2 - 1 is more conservative because it uses a smaller degrees of freedom. (b) Using the same hypotheses and sample data, is the Pvalue smaller for larger degrees of freedom? How might a larger P value impact the significance of a test? The Pvalue will be larger. Using a larger Pvalue might lead to concluding that a test is significant. The Pvalue will be smaller. Using a smaller Pvalue might lead to concluding that a test is not significant. The Pvalue will be smaller. Using a smaller Pvalue might lead to concluding that a test is significant. The Pvalue will be larger. Using a larger Pvalue might lead to concluding that a test is not significant. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 20/26 10/29/2015 Week 6 Assignment 31.-/1 pointsBBUnderStat11 8.5.007. When conducting a test for the difference of means for two independent populations x1 and x2, what alternate hypothesis would indicate that the mean of the x2 population is smaller than that of the x1 population? Express the alternate hypothesis in two ways. H1: 1 = 2 or H1: 1 - 2 = 0 H1: 1 > 2 or H1: 1 - 2 > 0 H1: 1 > 2 or H1: 2 - 1 > 0 H1: 1 < 2 or H1: 1 - 2 < 0 https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 21/26 10/29/2015 Week 6 Assignment 32.-/8 pointsBBUnderStat11 8.5.015. REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and nonREM sleep. However, it is thought that children have more REM sleep than adults. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 10 children (9 years old) showed that they had an average REM sleep time of x1 = 2.7 hours per night. From previous studies, it is known that 1 = 0.8 hour. Another random sample of n2 = 10 adults showed that they had an average REM sleep time of x2 = 1.80 hours per night. Previous studies show that 2 = 0.9 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance. (a) What is the level of significance? State the null and alternate hypotheses. H0: 1 = 2 H1: 1 2 H0: 1 < 2 H1: 1 = 2 H0: 1 = 2 H1: 1 < 2 H0: 1 = 2 H1: 1 > 2 (b) What sampling distribution will you use? What assumptions are you making? The Student's t. We assume that both population distributions are approximately normal with known standard deviations. The standard normal. We assume that both population distributions are approximately normal with known standard deviations. The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. What is the value of the sample test statistic? (Test the difference 1 2. Round your answer to two decimal places.) (c) Find (or estimate) the Pvalue. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the Pvalue. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 22/26 10/29/2015 Week 6 Assignment (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? At the = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. (e) Interpret your conclusion in the context of the application. Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults. Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults. Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults. Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 23/26 10/29/2015 Week 6 Assignment 33.-/4 pointsBBUnderStat11 8.5.028. Consider independent random samples from two populations that are normal or approximately normal, or the case in which both sample sizes are at least 30. Then, if 1 and 2 are unknown but we have reason to believe that 1 = 2, we can pool the standard deviations. Using sample sizes n1 and n2, the sample test statistic x1 x2 has a Student's t distribution where x1 x2 t = s 1 1 + n n1 2 with degrees of freedom d.f. = n1 + n2 2 and where the pooled standard deviation s is s = (n1 1)s12 + (n2 1)s22 n1 + n2 2 Note: With statistical software, select the pooled variance or equal variance options. (a) There are many situations in which we want to compare means from populations having standard deviations that are equal. This method applies even if the standard deviations are known to be only approximately equal. Consider a report regarding average incidence of fox rabies in two regions. For region I, n1 = 14, x1 4.90, and s1 = 2.52 and for region II, n2 = 19, x2 3.85, and s2 = 2.31. The two sample standard deviations are sufficiently close that we can assume 1 = 2. Use the method of pooled standard deviation to consider the report, testing if there is a difference in population mean average incidence of rabies at the 1% level of significance. (Round your tvalue to three decimal places and your Pvalue to four decimal places.) t = Pvalue = Conclusion Reject the null hypothesis, there is insufficient evidence to show a difference in mean average incidence of rabies. Fail to reject the null hypothesis, there is insufficient evidence to show a difference in mean average incidence of rabies. Reject the null hypothesis, there is sufficient evidence to show a difference in mean average incidence of rabies. Fail to reject the null hypothesis, there is sufficient evidence to show a difference in mean average incidence of rabies. (b) Compare the t value calculated in part (a) using the pooled standard deviation with the t value calculated using the unpooled standard deviation. Compare the degrees of freedom for the sample test statistic. Compare the conclusions. The t values are very similar. The degrees of freedom are the same. The conclusions are the same. The t values are very similar. The degrees of freedom for the unpooled method is much larger. The conclusions are the same. The t value for the unpooled method is much larger. The degrees of freedom are the same. The conclusions are the same. The t values are very similar. The degrees of freedom are the same. We reject the null using unpooled s but fail to reject using pooled s. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 24/26 10/29/2015 Week 6 Assignment 34.-/8 pointsBBUnderStat11 8.5.029. Would you favor spending more federal tax money on the arts? Of a random sample of n1 = 229 women, r1 = 68 responded yes. Another random sample of n2 = 171 men showed that r2 = 61 responded yes. Does this information indicate a difference (either way) between the population proportion of women and the population proportion of men who favor spending more federal tax dollars on the arts? Use = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. H0: p1 = p2 H1: p1 > p2 H0: p1 = p2 H1: p1 < p2 H0: p1 < p2 H1: p1 = p2 H0: p1 = p2 H1: p1 p2 (b) What sampling distribution will you use? What assumptions are you making? The Student's t. The number of trials is sufficiently large. The standard normal. We assume the population distributions are approximately normal. The Student's t. We assume the population distributions are approximately normal. The standard normal. The number of trials is sufficiently large. What is the value of the sample test statistic? (Test the difference p1 p2. Do not use rounded values. Round your final answer to two decimal places.) (c) Find (or estimate) the Pvalue. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the Pvalue. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 25/26 10/29/2015 Week 6 Assignment (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. Fail to reject the null hypothesis, there is insufficient evidence that the proportion of women favoring more tax dollars for the arts is different from the proportion of men. Reject the null hypothesis, there is sufficient evidence that the proportion of women favoring more tax dollars for the arts is different from the proportion of men. Fail to reject the null hypothesis, there is sufficient evidence that the proportion of women favoring more tax dollars for the arts is different from the proportion of men. Reject the null hypothesis, there is insufficient evidence that the proportion of women favoring more tax dollars for the arts is different from the proportion of men. https://www.webassign.net/web/Student/Assignment-Responses/last?dep=12615421 26/26 \f