1) A new SAT study course is tested on 12 individuals. Pre-course and post-course scores are recorded. Of interest is the average increase in SAT scores. The following data is collected. Conduct a hypothesis test at the 5% level. Pre-course score Post-course score 1200 1320 940 920 1030 1130 840 880 1100 1070 1250 1320 860 860 1330 1370 790 770 990 1040 1110 1200 740 850 NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. H0: d < 0 H0: d 0 H0: d > 0 H0: d 0 Part (b) State the alternative hypothesis. Ha: d = 0 Ha: d > 0 Ha: d 0 Ha: d < 0 Part (c) In words, state what your random variable x d represents. The variable x d represents the sample mean difference in SAT scores before the course and after the course. The variable x d represents the population mean difference in SAT scores before the course and after the course. The variable x d represents the sample total difference in the scores before and after the course. The variable x d represents the population total difference in the scores before and after the course. Part (d) State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom.): ____ Part (e) What is the test statistic? (Round your answer to two decimal places.) = Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. If H0 is false, then there is a chance equal to the p-value that the sample average difference between the post-course scores and pre-course scores is less than 45.83. If H0 is true, then there is a chance equal to the p-value that the sample average difference between the post-course scores and pre-course scores is at least 45.83. If H0 is true, then there is a chance equal to the p-value that the sample average difference between the post-course scores and pre-course scores is less than 45.83. If H0 is false, then there is a chance equal to the p-value that the sample average difference between the post-course scores and pre-course scores is at least 45.83. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha: = (ii) Decision: reject the null hypothesis do not reject the null hypothesis (iii) Reason for decision: Since p-value > , we do not reject the null hypothesis. hypothesis. Since p-value < , we do not reject the null Since p-value > , we reject the null hypothesis. Since p-value < , we reject the null hypothesis. (iv) Conclusion: There is sufficient evidence to show that the average post-course SAT score is larger than the average pre-course SAT score. There is not sufficient evidence to show that the average post-course SAT score is larger than the average pre-course SAT score. Part (i) Explain how you determined which distribution to use. The t-distribution will be used because the samples are dependent. The standard normal distribution will be used because the samples are independent and the population standard deviation is known. The standard normal distribution will be used because the samples involve the difference in proportions. The t-distribution will be used because the samples are independent and the population standard deviation is not known. 2) A student at a four-year college claims that average enrollment at four-year colleges is higher than at two-year colleges in the United States. Two surveys are conducted. Of the 35 two-year colleges surveyed, the average enrollment was 5068 with a standard deviation of 4777. Of the 35 four-year colleges surveyed, the average enrollment was 5466 with a standard deviation of 8191. Conduct a hypothesis test at the 5% level. NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. H0: 2yr < 4yr H0: 2yr > 4yr H0: 2yr 4yr H0: 2yr 4yr Part (b) State the alternative hypothesis. Ha: 2yr 4yr Ha: 2yr > 4yr Ha: 2yr < 4yr Ha: 2yr = 4yr Part (c) In words, state what your random variable x 2yr x 4yr represents. x 2yr x 4yr 2yr x 4yr represents the difference in enrollment of two-year colleges and four- x 4yr represents the average difference in enrollment of two-year colleges represents the average enrollment in two-year colleges and four-year colleges. x year colleges. x 2yr and four-year colleges. x 2yr x 4yr represents the difference in the average enrollment of two-year colleges and four-year colleges. Part (d) State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom. Round your answer to two decimal places.): ______ Part (e) What is the test statistic? (Round your answer to two decimal places.) = Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. If H0 is false, then there is a chance equal to the p-value that the sample average enrollment at 4-year colleges is at least 398 less than the sample average enrollment at 2year colleges. If H0 is false, then there is a chance equal to the p-value that the sample average enrollment at 4-year colleges is at least 398 more than the sample average enrollment at 2year colleges. If H0 is true, then there is a chance equal to the p-value that the sample average enrollment at 4-year colleges is at least 398 less than the sample average enrollment at 2year colleges. If H0 is true, then there is a chance equal to the p-value that the sample average enrollment at 4-year colleges is at least 398 more than the sample average enrollment at 2year colleges. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha: = (ii) Decision: reject the null hypothesis do not reject the null hypothesis (iii) Reason for decision: Since p-value > , we reject the null hypothesis. Since p-value < , we do not reject the null hypothesis. Since p-value > , we do not reject the null hypothesis. Since p-value < , we reject the null hypothesis. (iv) Conclusion: There is sufficient evidence to show that the average enrollment at four-year colleges is higher than at two-year colleges in the United States. There is not sufficient evidence to show that the average enrollment at four-year colleges is higher than at two-year colleges in the United States. Part (i) Explain how you determined which distribution to use. The t-distribution will be used because the samples are dependent. The t-distribution will be used because the samples are independent and the population standard deviation is not known. The standard normal distribution will be used because the samples involve the difference in proportions. The standard normal distribution will be used because the samples are independent and the population standard deviation is known. 3) While her husband spent 2 hours picking out new speakers, a statistician decided to determine whether the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment. The population was Saturday afternoon shoppers. Out of 65 men, 23 said they enjoyed the activity. 7 of the 21 women surveyed claimed to enjoy the activity. Interpret the results of the survey. Conduct a hypothesis test at the 5% level. Let the subscript m = men and w = women. NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. H0: pm pw H0: pm > pw H0: pm pw H0: pm pw Part (b) State the alternative hypothesis. Ha: pm > pw Ha: pm = pw Ha: pm pw Ha: pm < pw Part (c) In words, state what your random variable P'm P'w represents. P'm P'w represents the difference in the average number of men who enjoy shopping for electronics and women who enjoy shopping for electronics. P'm P'w represents the difference between the number of men who enjoy shopping for electronics and women who enjoy shopping for electronics. P'm P'w represents the difference in the proportion of men who enjoy shopping for electronics and women who enjoy shopping for electronics. P'm P'w represents the proportion of people who enjoy shopping for electronics. Part (d) State the distribution to use for the test. (Round your answer to four decimal places.) P'm P'w ~___ ( ___ , ___ ) Part (e) What is the test statistic? (Round your answer to two decimal places.) = Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. If H0 is true, then there is a chance equal to the p-value that the proportion of men who enjoy shopping for electronic equipment is at least 0.0205 higher than the proportion of women who enjoy shopping for electronic equipment. If H0 is true, then there is a chance equal to the p-value that the proportion of men who enjoy shopping for electronic equipment is 0.0205 lower than the proportion of women who enjoy shopping for electronic equipment. If H0 is false, then there is a chance equal to the p-value that the proportion of men who enjoy shopping for electronic equipment is 0.0205 lower than the proportion of women who enjoy shopping for electronic equipment. If H0 is false, then there is a chance equal to the p-value that the proportion of men who enjoy shopping for electronic equipment is at least 0.0205 higher than the proportion of women who enjoy shopping for electronic equipment. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha: = (ii) Decision: reject the null hypothesis do not reject the null hypothesis (iii) Reason for decision: Since p-value > , we do not reject the null hypothesis. hypothesis. Since p-value < , we do not reject the null Since p-value < , we reject the null hypothesis. Since p-value > , we reject the null hypothesis. (iv) Conclusion: There is sufficient evidence to show that the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment. There is not sufficient evidence to show that the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment. 4) Marketing companies have collected data implying that teenage girls use more ring tones on their cellular phones than teenage boys do. In one particular study of 40 randomly chosen teenage girls and boys (20 of each) with cellular phones, the average number of ring tones for the girls was 3.1 with a standard deviation of 1.7. The average for the boys was 1.6 with a standard deviation of 0.8. Conduct a hypothesis test at the 5% level to determine if the averages are approximately the same or if the girls' average is higher than the boys' average. NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. H0: girls boys H0: girls < boys H0: girls boys H0: girls boys Part (b) State the alternative hypothesis. Ha: girls = boys Ha: girls boys Ha: girls > boys Ha: girls boys Part (c) In words, state what your random variable x girls x boys x girls x boys represents. represents the difference in the average number of ring tones for girls and boys. x girls x boys represents the average difference in the number of ring tones for girls x girls x boys represents the difference in the number of ring tones for girls and x girls x boys represents the average number of ring tones that girls and boys and boys. boys. have. Part (d) State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom. Round your answer to two decimal places.): _____ Part (e) What is the test statistic? (Round your answer to two decimal places.) = Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. If H0 is true, then there is a chance equal to the p-value that the sample average number of ring tones for girls is 1.5 less than the sample average number of ring tones for boys. If H0 is false, then there is a chance equal to the p-value that the sample average number of ring tones for girls is at least 1.5 more than the sample average number of ring tones for boys. If H0 is false, then there is a chance equal to the p-value that the sample average number of ring tones for girls is 1.5 less than the sample average number of ring tones for boys. If H0 is true, then there is a chance equal to the p-value that the sample average number of ring tones for girls is at least 1.5 more than the sample average number of ring tones for boys. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha: = (ii) Decision: reject the null hypothesis do not reject the null hypothesis (iii) Reason for decision: Since p-value < , we do not reject the null hypothesis. hypothesis. Since p-value < , we reject the null Since p-value > , we do not reject the null hypothesis. reject the null hypothesis. Since p-value > , we (iv) Conclusion: There is sufficient evidence to show that the girls' average is higher than the boys' average. There is not sufficient evidence to show that the girls' average is higher than the boys' average. Part (i) Explain how you determined which distribution to use. The t-distribution will be used because the samples are independent and the population standard deviation is not known. The standard normal distribution will be used because the samples are independent and the population standard deviation is known. be used because the samples involve the difference in proportions. The standard normal distribution will The t-distribution will be used because the samples are dependent. 5) A study was conducted by the U.S. Army to see if applying antiperspirant to soldiers' feet for a few days before a major hike would help cut down on the number of blisters soldiers had on their feet. In the experiment, for three nights before they went on a 13-mile hike, a group of 328 West Point cadets put an alcohol-based antiperspirant on their feet. A "control group" of 339 soldiers put a similar, but inactive, preparation on their feet. On the day of the hike, the temperature reached 83F. At the end of the hike, 21% of the soldiers who had used the antiperspirant and 48% of the control group had developed foot blisters. Conduct a hypothesis test at the 5% level to see if the percent of soldiers using the antiperspirant was significantly lower than the control group. NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. H0: pa pna H0: pa > pna H0: pa pna H0: pa < pna Part (b) State the alternative hypothesis. Ha: pa pna Ha: pa < pna Ha: pa = pna Ha: pa > pna Part (c) In words, state what your random variable P'a P'na represents. P'a P'na represents the difference in the proportion of soldiers in the experimental group and control group who had blisters on their feet. P'a P'na represents the difference in the average number of soldiers in the experimental group and the control group who had blisters on their feet. P'a P'na represents the difference between the number of soldiers with and without blisters in the experimental and control groups. P'a P'na represents the proportion of soldiers who had blisters on their feet. Part (d) State the distribution to use for the test. (Round your answers to four decimal places.) P'a P'na ~ ___ ( ___ , ___ ) Part (e) What is the test statistic? (Round your answer to two decimal places.) = Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. If H0 is false, then there is a chance equal to the p-value that the proportion of soldiers who used the antiperspirant and developed blisters was at least 0.27 more than the proportion of soldiers who developed blisters in the control group. If H0 is true, then there is a chance equal to the p-value that the proportion of soldiers who used the antiperspirant and developed blisters was at least 0.27 lower than the proportion of soldiers who developed blisters in the control group. If H0 is true, then there is a chance equal to the p-value that the proportion of soldiers who used the antiperspirant and developed blisters was at least 0.27 more than the proportion of soldiers who developed blisters in the control group. If H0 is false, then there is a chance equal to the p-value that the proportion of soldiers who used the antiperspirant and developed blisters was at least 0.27 lower than the proportion of soldiers who developed blisters in the control group. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha: = (ii) Decision: reject the null hypothesis do not reject the null hypothesis (iii) Reason for decision: Since p-value < , we do not reject the null hypothesis. hypothesis. Since p-value < , we reject the null Since p-value > , we reject the null hypothesis. Since p-value > , we do not reject the null hypothesis. (iv) Conclusion: There is sufficient evidence to show that the percent of soldiers using the antiperspirant was significantly lower than the control group. There is not sufficient evidence to show that the percent of soldiers using the antiperspirant was significantly lower than the control group. Part (i) Explain how you determined which distribution to use. The standard normal distribution will be used because the samples involve the difference in proportion The standard normal distribution will be used because the samples are independent and the population standard deviation is known. The t-distribution will be used because the samples are independent and the population standard deviation is not known. The t-distribution will be used because the samples are dependent. 6) At Rachel's 11th birthday party, 8 girls were timed to see how long (in seconds) they could hold their breath in a relaxed position. After a two-minute rest, they timed themselves while jumping. The girls thought that the jumping would not affect their times, on average. Test their hypothesis at the 5% level. Relaxed time (seconds) Jumping time (seconds) 27 21 49 41 32 28 22 21 23 25 45 43 37 35 29 32 NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. H0: d = 0 H0: d 0 H0: d 0 H0: d 0 Part (b) State the alternative hypothesis. Ha: d > 0 Ha: d = 0 Ha: d < 0 Ha: d 0 Part (c) In words, state what your random variable x d x d represents. represents the difference in the average amount of time a girl can hold her breath while relaxed and while jumping. x d represents the amount of time each girl can hold her breath. x breath. d represents the average difference in the length of time each girl can hold her x d represents the difference in time for each girl to hold her breath while relaxed and while jumping. Part (d) State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom.): ____ Part (e) What is the test statistic? (Round your answer to two decimal places.) = Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. If H0 is true, then there is a chance equal to the p-value that the sample average difference between jumping times and relaxed times is between 2.25 and 2.25. If H0 is false, then there is a chance equal to the p-value that the sample average difference between jumping times and relaxed times is between 2.25 and 2.25. If H0 is true, then there is a chance equal to the p-value that the sample average difference between jumping times and relaxed times is 2.25 or less OR 2.25 or more. If H0 is false, then there is a chance equal to the p-value that the sample average difference between jumping times and relaxed times is 2.25 or less OR 2.25 or more. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha: = (ii) Decision: reject the null hypothesis do not reject the null hypothesis (iii) Reason for decision: Since p-value > , we reject the null hypothesis. hypothesis. Since p-value > , we do not reject the null Since p-value < , we do not reject the null hypothesis. Since p-value < , we reject the null hypothesis. (iv) Conclusion: There is sufficient evidence to show that the relaxed time, on average, is different than the jumping time. There is not sufficient evidence to show that the relaxed time, on average, is different than the jumping time. Part (i) Explain how you determined which distribution to use. The t-distribution will be used because the samples are dependent. The t-distribution will be used because the samples are independent and the population standard deviation is not known. The standard normal distribution will be used because the samples involve the difference in proportions. The standard normal distribution will be used because the samples are independent and the population standard deviation is known. 7) One of the questions in a study of marital satisfaction of dual-career couples was to rate the statement, "I'm pleased with the way we divide the responsibilities for childcare." The ratings went from 1 (strongly agree) to 5 (strongly disagree). Below are ten of the paired responses for husbands and wives. Conduct a hypothesis test at the 5% level to see if the average difference in the husband's versus the wife's satisfaction level is negative (meaning that, within the partnership, the husband is happier than the wife). Wife's score 3 4 2 3 4 2 1 1 2 4 Husband's score 2 1 1 3 2 1 1 1 2 4 NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. H0: d < 0 H0: d > 0 H0: d 0 H0: d 0 Part (b) State the alternative hypothesis. Ha: d 0 Ha: d = 0 Ha: d < 0 Ha: d > 0 Part (c) In words, state what your random variable x d represents. x d represents the average satisfaction level of husbands and wives. x d represents the average difference in the husbands' and the wives' satisfaction levels. x wives. d represents the difference in the average satisfaction level of husbands and x d represents the difference in the satisfaction levels of husbands and wives. Part (d) State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom.): ____ Part (e) What is the test statistic? (Round your answer to two decimal places.) = Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. If H0 is false, then there is a chance equal to the p-value that the sample average difference between the husbands' scores and the wives' scores is at most 0.8. If H0 is true, then there is a chance equal to the p-value that the sample average difference between the husbands' scores and the wives' scores is greater than 0.8. If H0 is false, then there is a chance equal to the p-value that the sample average difference between the husbands' scores and the wives' scores is greater than 0.8. If H0 is true, then there is a chance equal to the p-value that the sample average difference between the husbands' scores and the wives' scores is at most 0.8. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha: = (ii) Decision: reject the null hypothesis do not reject the null hypothesis (iii) Reason for decision: Since p-value > , we do not reject the null hypothesis. hypothesis. Since p-value < , we do not reject the null Since p-value < , we reject the null hypothesis. Since p-value > , we reject the null hypothesis. (iv) Conclusion: There is sufficient evidence to show that, on average, the husbands are more pleased than the wives with the division of childcare. There is not sufficient evidence to show that, on average, the husbands are more pleased than the wives with the division of childcare. Part (i) Explain how you determined which distribution to use. The standard normal distribution will be used because the samples are independent and the population standard deviation is known. The t-distribution will be used because the samples are independent and the population standard deviation is not known. are dependent. The t-distribution will be used because the samples The standard normal distribution will be used because the samples involve the difference in proportions. 8) Average entry level salaries for college graduates with mechanical engineering degrees and electrical engineering degrees are believed to be approximately the same. A recruiting office thinks that the average mechanical engineering salary is actually lower than the average electrical engineering salary. The recruiting office randomly surveys 44 entry level mechanical engineers and 52 entry level electrical engineers. Their average salaries were $46,000 and $46,700, respectively. Their standard deviations were $3410 and $4230, respectively. Conduct a hypothesis test at the 5% level to determine if you agree that the average entry level mechanical engineering salary is lower than the average entry level electrical engineering salary. Let the subscript m = mechanical and e = electrical. NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. H0: m e H0: m e H0: m < e H0: m e Part (b) State the alternative hypothesis. Ha: m e Ha: m e Ha: m = e Ha: m < e Part (c) In words, state what your random variable x m x e x m x e represents. represents the average starting salary of entry-level mechanical engineers and electrical engineers. x m x e represents the difference in starting salaries of entry-level mechanical engineers and electrical engineers. x m x e represents the average difference in the starting salaries of entry-level mechanical engineers and electrical engineers. x m x e represents the difference in the average starting salaries of entry-level mechanical engineers and electrical engineers. Part (d) State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom. Round your answer to two decimal places.): ____ Part (e) What is the test statistic? (Round your answer to two decimal places.) = Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. If H0 is true, then there is a chance equal to the p-value that the sample average salary of mechanical engineers is at least $700 less than the sample average salary of electrical engineers. If H0 is false, then there is a chance equal to the p-value that the sample average salary of mechanical engineers is at least $700 less than the sample average salary of electrical engineers. If H0 is false, then there is a chance equal to the p-value that the sample average salary of mechanical engineers is $700 more than the sample average salary of electrical engineers. If H0 is true, then there is a chance equal to the p-value that the sample average salary of mechanical engineers is $700 more than the sample average salary of electrical engineers. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion. (i) Alpha: = (ii) Decision: reject the null hypothesis do not reject the null hypothesis (iii) Reason for decision: Since p-value < , we reject the null hypothesis. hypothesis. Since p-value < , we do not reject the null Since p-value > , we do not reject the null hypothesis. Since p-value > , we reject the null hypothesis. (iv) Conclusion: There is sufficient evidence to show that the average entry level mechanical engineering salary is lower than the average entry level electrical engineering salary. There is not sufficient evidence to show that the average entry level mechanical engineering salary is lower than the average entry level electrical engineering salary. Part (i) Explain how you determined which distribution to use. The t-distribution will be used because the samples are independent and the population standard deviation is not known. The t-distribution will be used because the samples are dependent. The standard normal distribution will be used because the samples involve the difference in proportions. The standard normal distribution will be used because the samples are independent and the population standard deviation is known