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Exam 10: B: Inference From Small Samples

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A researcher believes she has designed a keyboard that is more efficient to use than a standard keyboard. In order to help decide if this is the case, typing speeds were taken for eight different people on each keyboard, new and standard original. The lengths of time, in minutes, for each of the people to type a preselected manuscript are listed below. Assume the two population distributions are normal. Use the data to determine if the original keyboard yields slower times. Use a significance level of α\alpha = 0.05. A researcher believes she has designed a keyboard that is more efficient to use than a standard keyboard. In order to help decide if this is the case, typing speeds were taken for eight different people on each keyboard, new and standard original. The lengths of time, in minutes, for each of the people to type a preselected manuscript are listed below. Assume the two population distributions are normal. Use the data to determine if the original keyboard yields slower times. Use a significance level of \alpha = 0.05.

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Let Original keyboard be population 1, New keyboard be population 2, and Let Original keyboard be population 1, New keyboard be population 2, and be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8. The sample mean and sample standard deviation of the differences, d, are calculated as follows: = 25/8 = 3.125. = = 2.532. Hence, the test statistic is = 3.125/0.8952 = 3.491. Reject if t > = 1.895. Since t > 1.895, reject , and conclude that the original keyboard yields slower times. be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Let Original keyboard be population 1, New keyboard be population 2, and be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8. The sample mean and sample standard deviation of the differences, d, are calculated as follows: = 25/8 = 3.125. = = 2.532. Hence, the test statistic is = 3.125/0.8952 = 3.491. Reject if t > = 1.895. Since t > 1.895, reject , and conclude that the original keyboard yields slower times. Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8.
The sample mean and sample standard deviation of the differences, d, are calculated as follows: Let Original keyboard be population 1, New keyboard be population 2, and be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8. The sample mean and sample standard deviation of the differences, d, are calculated as follows: = 25/8 = 3.125. = = 2.532. Hence, the test statistic is = 3.125/0.8952 = 3.491. Reject if t > = 1.895. Since t > 1.895, reject , and conclude that the original keyboard yields slower times. = 25/8 = 3.125. Let Original keyboard be population 1, New keyboard be population 2, and be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8. The sample mean and sample standard deviation of the differences, d, are calculated as follows: = 25/8 = 3.125. = = 2.532. Hence, the test statistic is = 3.125/0.8952 = 3.491. Reject if t > = 1.895. Since t > 1.895, reject , and conclude that the original keyboard yields slower times. = Let Original keyboard be population 1, New keyboard be population 2, and be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8. The sample mean and sample standard deviation of the differences, d, are calculated as follows: = 25/8 = 3.125. = = 2.532. Hence, the test statistic is = 3.125/0.8952 = 3.491. Reject if t > = 1.895. Since t > 1.895, reject , and conclude that the original keyboard yields slower times. = 2.532. Hence, the test statistic is Let Original keyboard be population 1, New keyboard be population 2, and be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8. The sample mean and sample standard deviation of the differences, d, are calculated as follows: = 25/8 = 3.125. = = 2.532. Hence, the test statistic is = 3.125/0.8952 = 3.491. Reject if t > = 1.895. Since t > 1.895, reject , and conclude that the original keyboard yields slower times. = 3.125/0.8952 = 3.491. Reject Let Original keyboard be population 1, New keyboard be population 2, and be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8. The sample mean and sample standard deviation of the differences, d, are calculated as follows: = 25/8 = 3.125. = = 2.532. Hence, the test statistic is = 3.125/0.8952 = 3.491. Reject if t > = 1.895. Since t > 1.895, reject , and conclude that the original keyboard yields slower times. if t > Let Original keyboard be population 1, New keyboard be population 2, and be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8. The sample mean and sample standard deviation of the differences, d, are calculated as follows: = 25/8 = 3.125. = = 2.532. Hence, the test statistic is = 3.125/0.8952 = 3.491. Reject if t > = 1.895. Since t > 1.895, reject , and conclude that the original keyboard yields slower times. = 1.895. Since t > 1.895, reject Let Original keyboard be population 1, New keyboard be population 2, and be the true average difference in typing speed for the two keyboards. The hypotheses to be tested are Differences: d = (Speed of original keyboard - Speed of new keyboard) are d =3, 1, 2, 2, 4, 0, 5, and 8. The sample mean and sample standard deviation of the differences, d, are calculated as follows: = 25/8 = 3.125. = = 2.532. Hence, the test statistic is = 3.125/0.8952 = 3.491. Reject if t > = 1.895. Since t > 1.895, reject , and conclude that the original keyboard yields slower times. , and conclude that the original keyboard yields slower times.

Assume that the population distributions of times (in hours) of two different surgeries are normal with equal variances. Two random samples, drawn independently from the populations, showed the following statistics. Assume that the population distributions of times (in hours) of two different surgeries are normal with equal variances. Two random samples, drawn independently from the populations, showed the following statistics. = 10, = 2.5, = 0.04 = 11, = 2.6, = 0.09 Construct and interpret a 90% confidence interval for the true difference in mean amount of time of the two surgeries. = 10, Assume that the population distributions of times (in hours) of two different surgeries are normal with equal variances. Two random samples, drawn independently from the populations, showed the following statistics. = 10, = 2.5, = 0.04 = 11, = 2.6, = 0.09 Construct and interpret a 90% confidence interval for the true difference in mean amount of time of the two surgeries. = 2.5, Assume that the population distributions of times (in hours) of two different surgeries are normal with equal variances. Two random samples, drawn independently from the populations, showed the following statistics. = 10, = 2.5, = 0.04 = 11, = 2.6, = 0.09 Construct and interpret a 90% confidence interval for the true difference in mean amount of time of the two surgeries. = 0.04 Assume that the population distributions of times (in hours) of two different surgeries are normal with equal variances. Two random samples, drawn independently from the populations, showed the following statistics. = 10, = 2.5, = 0.04 = 11, = 2.6, = 0.09 Construct and interpret a 90% confidence interval for the true difference in mean amount of time of the two surgeries. = 11, Assume that the population distributions of times (in hours) of two different surgeries are normal with equal variances. Two random samples, drawn independently from the populations, showed the following statistics. = 10, = 2.5, = 0.04 = 11, = 2.6, = 0.09 Construct and interpret a 90% confidence interval for the true difference in mean amount of time of the two surgeries. = 2.6, Assume that the population distributions of times (in hours) of two different surgeries are normal with equal variances. Two random samples, drawn independently from the populations, showed the following statistics. = 10, = 2.5, = 0.04 = 11, = 2.6, = 0.09 Construct and interpret a 90% confidence interval for the true difference in mean amount of time of the two surgeries. = 0.09 Construct and interpret a 90% confidence interval for the true difference in mean amount of time of the two surgeries.

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The pooled estimate of the common variance is The pooled estimate of the common variance is = (0.36 + 0.90)/19 = 0.0663. Then, the 90% confidence interval is = (2.5 - 2.6) 1.729 (0.1125) = -0.10 0.1945, or -0.2945 < < 0.0945. Since this interval contains 0, the sample evidence supports that the two surgeries, on average, do not take significantly different amounts of time. = (0.36 + 0.90)/19 = 0.0663. Then, the 90% confidence interval is The pooled estimate of the common variance is = (0.36 + 0.90)/19 = 0.0663. Then, the 90% confidence interval is = (2.5 - 2.6) 1.729 (0.1125) = -0.10 0.1945, or -0.2945 < < 0.0945. Since this interval contains 0, the sample evidence supports that the two surgeries, on average, do not take significantly different amounts of time. = (2.5 - 2.6) The pooled estimate of the common variance is = (0.36 + 0.90)/19 = 0.0663. Then, the 90% confidence interval is = (2.5 - 2.6) 1.729 (0.1125) = -0.10 0.1945, or -0.2945 < < 0.0945. Since this interval contains 0, the sample evidence supports that the two surgeries, on average, do not take significantly different amounts of time. 1.729 (0.1125) = -0.10 The pooled estimate of the common variance is = (0.36 + 0.90)/19 = 0.0663. Then, the 90% confidence interval is = (2.5 - 2.6) 1.729 (0.1125) = -0.10 0.1945, or -0.2945 < < 0.0945. Since this interval contains 0, the sample evidence supports that the two surgeries, on average, do not take significantly different amounts of time. 0.1945, or -0.2945 < The pooled estimate of the common variance is = (0.36 + 0.90)/19 = 0.0663. Then, the 90% confidence interval is = (2.5 - 2.6) 1.729 (0.1125) = -0.10 0.1945, or -0.2945 < < 0.0945. Since this interval contains 0, the sample evidence supports that the two surgeries, on average, do not take significantly different amounts of time. < 0.0945. Since this interval contains 0, the sample evidence supports that the two surgeries, on average, do not take significantly different amounts of time.

Snowmobile Speeds Narrative A customer was interested in comparing the top speed (in kilometres per hour) of two models of snowmobiles. The customer selected two independent random samples of the snowmobiles and calculated the following summary information: Snowmobile Speeds Narrative A customer was interested in comparing the top speed (in kilometres per hour) of two models of snowmobiles. The customer selected two independent random samples of the snowmobiles and calculated the following summary information: Assume the distribution of top speeds is approximately normal. -Refer to Snowmobile Speeds Narrative. Set up the appropriate rejection region for the hypotheses above assuming = 0.05. Assume the distribution of top speeds is approximately normal. -Refer to Snowmobile Speeds Narrative. Set up the appropriate rejection region for the hypotheses above assuming Snowmobile Speeds Narrative A customer was interested in comparing the top speed (in kilometres per hour) of two models of snowmobiles. The customer selected two independent random samples of the snowmobiles and calculated the following summary information: Assume the distribution of top speeds is approximately normal. -Refer to Snowmobile Speeds Narrative. Set up the appropriate rejection region for the hypotheses above assuming = 0.05. = 0.05.

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Reject Reject if | t | > = 2.131. if | t | > Reject if | t | > = 2.131. = 2.131.

Motorcycle Fuel Consumption Narrative A Harley Davidson dealer wants to know the average fuel consumption (in litres per 100 km) of a 1992 XLT. A random sample of 17 was taken from a normally distributed population and produced a mean of 4.5 L/100 km and a standard deviation of 0.36 L/100 km. -Refer to Motorcycle Fuel Consumption Narrative. Interpret the interval in the previous question.

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Question 4

Bottling Productivity Narrative Five soft drink bottling companies have agreed to implement a time management program in hopes of increasing productivity (measured in cases of soft drinks bottled per hour). The number of cases of soft drinks bottled per hour before and after the implementation of the program are listed below: Company Bottling Productivity Narrative Five soft drink bottling companies have agreed to implement a time management program in hopes of increasing productivity (measured in cases of soft drinks bottled per hour). The number of cases of soft drinks bottled per hour before and after the implementation of the program are listed below: Company -Refer to Bottling Productivity Narrative. Find the approximate p-value. -Refer to Bottling Productivity Narrative. Find the approximate p-value.

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Question 5

Disinfectant Experiments Narrative An experiment to determine the efficacy of using 95% ethanol or 20% bleach as a disinfectant in removing bacterial and fungal contamination when culturing plant tissues was repeated 15 times for each disinfectant. The plant tissue being cultured was sweet potato: Five cuttings per plant were placed on a petri dish for each disinfectant and stored at 25°C for four weeks. The observation reported was the number of uncontaminated eggplant cuttings after the four-week storage. Disinfectant Disinfectant Experiments Narrative An experiment to determine the efficacy of using 95% ethanol or 20% bleach as a disinfectant in removing bacterial and fungal contamination when culturing plant tissues was repeated 15 times for each disinfectant. The plant tissue being cultured was sweet potato: Five cuttings per plant were placed on a petri dish for each disinfectant and stored at 25°C for four weeks. The observation reported was the number of uncontaminated eggplant cuttings after the four-week storage. Disinfectant -Refer to Disinfectant Experiments Narrative. Using the information from the previous question, are you willing to conclude that there is a significant difference in the mean numbers of uncontaminated eggplants for the two disinfectants tested? -Refer to Disinfectant Experiments Narrative. Using the information from the previous question, are you willing to conclude that there is a significant difference in the mean numbers of uncontaminated eggplants for the two disinfectants tested?

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Question 6

Running Shoes Narrative A new runner has decided to purchase a new pair of running shoes. He has narrowed his choices to two brands, each of which would be appropriate for his use. His concern is whether there is a significant difference in the average wear between the two brands of shoes. He enlists a random sample of six veteran runners to test the shoes. Each runner wore each brand of shoe until it wore out. The following data were recorded, representing the number of weeks each runner used each pair of shoes: Runner Running Shoes Narrative A new runner has decided to purchase a new pair of running shoes. He has narrowed his choices to two brands, each of which would be appropriate for his use. His concern is whether there is a significant difference in the average wear between the two brands of shoes. He enlists a random sample of six veteran runners to test the shoes. Each runner wore each brand of shoe until it wore out. The following data were recorded, representing the number of weeks each runner used each pair of shoes: Runner -Refer to Running Shoes Narrative. Perform the appropriate test of hypothesis to determine whether there is a significant difference in the average wear between the two brands of shoes. Use the 5% level of significance. -Refer to Running Shoes Narrative. Perform the appropriate test of hypothesis to determine whether there is a significant difference in the average wear between the two brands of shoes. Use the 5% level of significance.

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Question 7

Snowmobile Speeds Narrative A customer was interested in comparing the top speed (in kilometres per hour) of two models of snowmobiles. The customer selected two independent random samples of the snowmobiles and calculated the following summary information: Snowmobile Speeds Narrative A customer was interested in comparing the top speed (in kilometres per hour) of two models of snowmobiles. The customer selected two independent random samples of the snowmobiles and calculated the following summary information: Assume the distribution of top speeds is approximately normal. -Refer to Snowmobile Speeds Narrative. What is the appropriate conclusion? Be sure to justify your answer. Assume the distribution of top speeds is approximately normal. -Refer to Snowmobile Speeds Narrative. What is the appropriate conclusion? Be sure to justify your answer.

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Question 8

Manufacturing Garment Average Narrative A garment manufacturing company recorded the amount of time that it took to make a pair of jeans on eight different occasions. The times in minutes are as follows: 12.5, 13.0, 11.9, 10.2, 13.1, 13.6, 13.8, and 14.0. Assume these measurements were taken from a population with a normal distribution. Do the sample data suggest that the average time it takes this company to make a pair of jeans is less than 13.5 minutes? -Refer to Manufacturing Garment Average Narrative. State the appropriate hypotheses.

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Question 9

Given a random variable that has a t distribution with the specified degrees of freedom, in each of the following cases what percentage of the time will its value fall in the indicated region? a. 15 degrees of freedom, between -2.131 and 2.131 b. 19 degrees of freedom, between -2.539 and 2.539 c. 23 degrees of freedom, between -1.319 and 1.319 d. 10 degrees of freedom, between -3.169 and 3.169

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Question 10

Average Telephone Time on Hold Narrative A customer service representative was interested in comparing the average time (in minutes) customers are placed on hold when calling Gaz Metropolitain and Hydro-Quebec, both in Quebec. The representative obtained two independent random samples and calculated the following summary information: Average Telephone Time on Hold Narrative A customer service representative was interested in comparing the average time (in minutes) customers are placed on hold when calling Gaz Metropolitain and Hydro-Quebec, both in Quebec. The representative obtained two independent random samples and calculated the following summary information: Assume the distributions of time a customer is on hold are approximately normal. -Refer to Average Telephone Time on Hold Narrative. Calculate the value of the test statistic. Assume the distributions of time a customer is on hold are approximately normal. -Refer to Average Telephone Time on Hold Narrative. Calculate the value of the test statistic.

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Question 11

Preservatives Narrative A food processing plant wants to compare two preservatives for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with preservative A and 16 are treated with preservative B. The number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are summarized in the table below. Preservatives Narrative A food processing plant wants to compare two preservatives for their effects on retarding spoilage. Suppose 16 cuts of fresh meat are treated with preservative A and 16 are treated with preservative B. The number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are summarized in the table below. -Refer to Preservatives Narrative. Calculate the pooled variance and the value of the test statistic. -Refer to Preservatives Narrative. Calculate the pooled variance and the value of the test statistic.

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Question 12

Studying Time Narrative A faculty advisor was interested in determining whether there is a difference between male and female students in the amount of time (in hours) spent studying on weeknights (Monday through Thursday). The advisor selected a random sample of 12 female students and a second random, but independent, sample of 10 male students and asked each student to indicate the average amount of time spent studying on a weeknight. The following summary statistics are obtained. Studying Time Narrative A faculty advisor was interested in determining whether there is a difference between male and female students in the amount of time (in hours) spent studying on weeknights (Monday through Thursday). The advisor selected a random sample of 12 female students and a second random, but independent, sample of 10 male students and asked each student to indicate the average amount of time spent studying on a weeknight. The following summary statistics are obtained. -Refer to Studying Time Narrative. Develop a 95% confidence interval for the average amount of time spent studying on weeknights by females. -Refer to Studying Time Narrative. Develop a 95% confidence interval for the average amount of time spent studying on weeknights by females.

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Question 13

Interest Rates Narrative The following 10 observations are interest rates on unpaid balances on credit cards for a department store: 10.4, 10.1, 9.5, 10.5, 10.6, 9.3, 9.9, 10.7, 9.5, and 10.0. -Refer to Interest Rates Narrative. Find the mean and standard deviation of these data.

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Question 14

Studying Time Narrative A faculty advisor was interested in determining whether there is a difference between male and female students in the amount of time (in hours) spent studying on weeknights (Monday through Thursday). The advisor selected a random sample of 12 female students and a second random, but independent, sample of 10 male students and asked each student to indicate the average amount of time spent studying on a weeknight. The following summary statistics are obtained. Studying Time Narrative A faculty advisor was interested in determining whether there is a difference between male and female students in the amount of time (in hours) spent studying on weeknights (Monday through Thursday). The advisor selected a random sample of 12 female students and a second random, but independent, sample of 10 male students and asked each student to indicate the average amount of time spent studying on a weeknight. The following summary statistics are obtained. -Refer to Studying Time Narrative. Perform the appropriate test of hypothesis to determine whether there is a significant difference between male and female students in average time spent studying on weeknights. Use = 0.05. -Refer to Studying Time Narrative. Perform the appropriate test of hypothesis to determine whether there is a significant difference between male and female students in average time spent studying on weeknights. Use Studying Time Narrative A faculty advisor was interested in determining whether there is a difference between male and female students in the amount of time (in hours) spent studying on weeknights (Monday through Thursday). The advisor selected a random sample of 12 female students and a second random, but independent, sample of 10 male students and asked each student to indicate the average amount of time spent studying on a weeknight. The following summary statistics are obtained. -Refer to Studying Time Narrative. Perform the appropriate test of hypothesis to determine whether there is a significant difference between male and female students in average time spent studying on weeknights. Use = 0.05. = 0.05.

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Question 15

Average Telephone Time on Hold Narrative A customer service representative was interested in comparing the average time (in minutes) customers are placed on hold when calling Gaz Metropolitain and Hydro-Quebec, both in Quebec. The representative obtained two independent random samples and calculated the following summary information: Average Telephone Time on Hold Narrative A customer service representative was interested in comparing the average time (in minutes) customers are placed on hold when calling Gaz Metropolitain and Hydro-Quebec, both in Quebec. The representative obtained two independent random samples and calculated the following summary information: Assume the distributions of time a customer is on hold are approximately normal. -Refer to Average Telephone Time on Hold Narrative. Set up the appropriate rejection region for the hypotheses above, assuming = 0.10. Assume the distributions of time a customer is on hold are approximately normal. -Refer to Average Telephone Time on Hold Narrative. Set up the appropriate rejection region for the hypotheses above, assuming Average Telephone Time on Hold Narrative A customer service representative was interested in comparing the average time (in minutes) customers are placed on hold when calling Gaz Metropolitain and Hydro-Quebec, both in Quebec. The representative obtained two independent random samples and calculated the following summary information: Assume the distributions of time a customer is on hold are approximately normal. -Refer to Average Telephone Time on Hold Narrative. Set up the appropriate rejection region for the hypotheses above, assuming = 0.10. = 0.10.

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Question 16

Earthquake Analysis Narrative The length of duration, in minutes, of earthquakes in British Columbia has been recorded for future analysis and information. The length of duration of a random sample of six earthquakes is as follows: 1.1, 0.9, 1.5, 0.7, 1.4, and 1.3. -Refer to Earthquake Analysis Narrative. Assuming the distribution of the length of duration of the earthquakes is approximately normal, find a 98% confidence interval for the true average duration of earthquakes in British Columbia.

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Question 17

Vertical Blinds Installation Narrative A drapery store manager was interested in determining whether a new employee can install vertical blinds faster than an employee who has been with the company for two years. The manager takes independent samples of ten vertical blind installations of each of the two employees and computes the following information. Vertical Blinds Installation Narrative A drapery store manager was interested in determining whether a new employee can install vertical blinds faster than an employee who has been with the company for two years. The manager takes independent samples of ten vertical blind installations of each of the two employees and computes the following information. -Refer to Vertical Blinds Installation Narrative. Set up the appropriate rejection region for the hypotheses above and assume \alpha = 0.05. -Refer to Vertical Blinds Installation Narrative. Set up the appropriate rejection region for the hypotheses above and assume α\alpha = 0.05.

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Question 18

Vertical Blinds Installation Narrative A drapery store manager was interested in determining whether a new employee can install vertical blinds faster than an employee who has been with the company for two years. The manager takes independent samples of ten vertical blind installations of each of the two employees and computes the following information. Vertical Blinds Installation Narrative A drapery store manager was interested in determining whether a new employee can install vertical blinds faster than an employee who has been with the company for two years. The manager takes independent samples of ten vertical blind installations of each of the two employees and computes the following information. -A logger knows the average time for his cutting machine to cut 20 trees is 9.8 minutes. A new machine on the market claims to cut the trees in less than 9.8 minutes. A random sample of 25 test runs on the new machine yielded a mean of 8.5 minutes with a standard deviation of 1.5. Do the sample data suggest the new machine cuts faster than the logger's machine? Test at the \alpha = 0.05 level. Assume the cutting time is normally distributed and interpret your results. -A logger knows the average time for his cutting machine to cut 20 trees is 9.8 minutes. A new machine on the market claims to cut the trees in less than 9.8 minutes. A random sample of 25 test runs on the new machine yielded a mean of 8.5 minutes with a standard deviation of 1.5. Do the sample data suggest the new machine cuts faster than the logger's machine? Test at the α\alpha = 0.05 level. Assume the cutting time is normally distributed and interpret your results.

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Question 19

Temperature Average Narrative The average low temperature for Victoria, B.C. in September is 12°C. The average low temperature for each of eight randomly selected years is 11.0, 12.4, 11.8, 10.9, 11.4, 12.2, 10.8, and 12.2. Assume the September low temperature distribution is normal. Do the sample data suggest the average low temperature is lower than 12°C? -Refer to Temperature Average Narrative. State the appropriate hypotheses.

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