1. Suppose that an accounting firm does a study to determine the time needed to complete one person's tax forms. It randomly surveys 200 people. The sample average is 23.1 hours. There is a known population standard deviation of 6.2 hours. The population distribution is assumed to be normal. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) (i) x = (ii) = (iii) x = (rounded to three decimal places) (iv) n= (v) n1= Part (b) Define the Random Variables X and x in words. X is the number of tax forms that an accounting firm completes, and x is the average number of tax forms that an accounting firm completes. X is the time needed to complete a person's tax forms, and x is the average time needed to complete 200 tax forms. x is the number of tax forms that an accounting firm completes, and X is the average number of tax forms that an accounting firm completes. x is the time needed to complete a person's tax forms, and X is the average time needed to complete 200 tax forms. Part (c) Which distribution should you use for this problem? (Round your answers to two decimal places.) X ~ __ ( __ , __ ) Explain your choice. The standard normal distribution should be used because the population standard deviation is known. The standard normal distribution should be used because the mean is given. The Student's t-distribution should be used because the sample standard deviation is given. The Student's t-distribution should be used because the sample mean is smaller than 30. Part (d) Construct a 90% confidence interval for the population average time to complete the tax forms. (i) State the confidence interval. (Round your answers to two decimal places.) ( __ , __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to two decimal places.) Part (e) If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what change should it make? It should increase the number of people surveyed. It should decrease the number of people surveyed. Part (f) If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Why? The level of confidence would be smaller because we have collected a smaller sample, obtaining less accurate information. The level of confidence would be larger because we have collected a smaller sample, obtaining less accurate information. There would be no change. Part (g) Suppose that the firm decided that it needed to be at least 96% confident of the population average length of time to within 1 hour. How would the number of people the firm surveys change? Why? The number of people surveyed would increase because more accurate information requires a larger sample. There would be no change. The number of people surveyed would decrease because more accurate information requires a smaller sample. 2. Suppose that insurance companies did a survey. They randomly surveyed 450 drivers and found that 320 claimed to always buckle up. We are interested in the population proportion of drivers who claim to always buckle up. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) (i) x= (ii) n= (iii) p' = (rounded to four decimal places) Part (b) Define the Random Variables X and P', in words. X is the number of people who do not buckle up, and P' is the proportion of people in the sample who do not buckle up. X is the number of people who claim they buckle up, and P' is the proportion of people in the sample who buckle up. X is the proportion of people in the sample who claim they buckle up, and P' is the number of people who buckle up. X is the proportion of people in the sample who do not buckle up, and P' is the number of people who do not buckle up. Part (c) Which distribution should you use for this problem? (Round your answer to four decimal places.) P' ~__ ( __ , __ ) Explain your choice. The Student's t-distribution should be used because we do not know the standard deviation. The normal distribution should be used because we are interested in proportions and the sample size is large. The binomial distribution should be used because there are two outcomes, buckle up or do not buckle up. The Student's t-distribution should be used because 10, which implies a small sample. np q Part (d) Construct a 95% confidence interval for the population proportion that claim to always buckle up. (i) State the confidence interval. (Round your answers to four decimal places.) ( __ , __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to four decimal places.) Part (e) If this survey were done by telephone, select 3 difficulties the companies might have in obtaining random results. Individuals may choose to participate or not participate in the phone survey. The individuals in the sample may not accurately reflect the population . Only people over the age of 24 would be included in the survey. Individuals may not tell the truth about buckling up. Children who do not answer the phone will not be included. 3. Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its average number of unoccupied seats per flight over the past year. To accomplish this, the records of 300 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is 11.7 seats and the sample standard deviation is 4.5 seats. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) (i) x = (ii) sx = (iii) n= (iv) n1= Part (b) Define the Random Variables X and x x , in words. is the number of unoccupied seats on a flight, and X is the average number of unoccupied seats for a sample of 300 flights. X is the number of full flights, and x is the average number of full flights for a sample of 300 flights. x is the number of full flights, and X is the average number of full flights for a sample of 300 flights. X is the number of unoccupied seats on a flight, and x is the average number of unoccupied seats for a sample of 300 flights. Part (c) Which distribution should you use for this problem? _____ (Enter your answer in the form z or tdf where df is the degrees of freedom.) Explain your choice. The Student's t-distribution should be used because the population standard deviation is not given. The standard normal distribution should be used because the sample standard deviation is known. The standard normal distribution should be used because the sample size is very large. The normal distribution should be used because we are interested in proportions and the sample size is large. Part (d) Construct a 92% confidence interval for the population average number of unoccupied seats per flight. (i) State the confidence interval. (Round your answers to two decimal places.) ( __ , __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to two decimal places.) 4. According to a recent survey of 1900 people, 62% feel that the president is doing an acceptable job. We are interested in the population proportion of people who feel the president is doing an acceptable job. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) Define the Random Variables X and P', in words. X is the number of people who feel the president is not doing an acceptable job, and P' is the proportion of people in the sample who feel the president is not doing an acceptable job. X is the number of people who feel the president is doing an acceptable job, and P' is the proportion of people in the sample who feel the president is doing an acceptable job. X is the proportion of people in the sample who feel the president is doing an acceptable job, and P' is the number of people who feel the president is doing an acceptable job. X is the proportion of people in the sample who feel the president is not doing an acceptable job, and P' is the number of people who feel the president is not doing an acceptable job. Part (b) Which distribution should you use for this problem? (Round your answers to four decimal places.) P' ~___ ( __ , __ ) Explain your choice. The Student's t-distribution should be used because we do not know the standard deviation. The standard normal distribution should be used np q because 10, which implies a large sample. The standard normal distribution should be used because we are interested in proportions and the sample size is large. The binomial distribution should be used because the two outcomes are "the president is doing a good job" and "the president is not doing a good job." Part (c) Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job. (i) State the confidence interval. (Round your answers to four decimal places.) ( __, __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to four decimal places.) 5. An article regarding interracial dating and marriage recently appeared in a newspaper. Of the 1708 randomly selected adults, 311 identified themselves as Latinos, 321 identified themselves as blacks, 255 identified themselves as Asians, and 778 identified themselves as whites. In this survey, 86% of blacks said that their families would welcome a white person into their families. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) We are interested in finding the 95% confidence interval for the percent of black families that would welcome a white person into their families. Define the Random Variables X and P', in words. X is the number of black families that would welcome a white person into their family, and P' is the proportion of black families that would welcome a white person into their family. X is the proportion of families that approve of interracial dating and marriage, and P' is the number of families that approve of interracial dating and marriage. X is the number of families that approve of interracial dating and marriage, and P' is the proportion of families that approve of interracial dating and marriage. X is the proportion of black families that would welcome a white person into their family, and P' is the number of black families that would welcome a white person into their family. Part (b) Which distribution should you use for this problem? (Round your answers to four decimal places.) P' ~ __ ( __ , __ ) Explain your choice. The binomial distribution should be used because the two outcomes are "welcome a white person into your home" and "do not welcome a white person into your home." The standard normal distribution should be used because we are interested in proportions and the sample size is large. The Student's t-distribution should be used because we do not know the standard deviation. The Student's t-distribution should be used because np q 10, which implies a small sample. Part (c) Construct a 95% confidence interval for the population proportion of blacks that would welcome a white person into their family. (i) State the confidence interval. (Round your answers to four decimal places.) ( __ , __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to four decimal places.) 6. An article regarding interracial dating and marriage recently appeared in a newspaper. Of the 1709 randomly selected adults, 315 identified themselves as Latinos, 323 identified themselves as blacks, 254 identified themselves as Asians, and 779 identified themselves as whites. Among Asians, 79% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) Construct the 95% confidence intervals for the three Asian responses. (Round your answers to four decimal places.) welcome a white person ( __ , __ ) welcome a Latino (__ , __ ) welcome a black person ( __ , __ ) Part (b) Even though the three point estimates are different, do any of the confidence intervals overlap? Which? (Select all that apply.) Yes, the intervals for Latinos and blacks overlap. Yes, the intervals for whites and blacks overlap. No confidence intervals overlap. Yes, the intervals for whites and Latinos overlap. Yes, all three intervals overlap. Part (c) For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions? The confidence intervals for white and Latino overlap, as do those for black and Latino, which means that there is no significant difference in their proportions. The confidence intervals for black and Latino overlap, and the overlapping region contains the black and Latino sample proportions; therefore, there is no evidence of a significant difference between their true proportions. The confidence intervals for white and Latino overlap, but the overlapping region does not contain the white or Latino sample proportions; therefore, there is evidence of a significant difference between their true proportions. Intervals that overlap do not give any indication about the difference of the true population proportion. There are no confidence intervals that overlap, implying that there is a significant difference in their population proportions. Part (d) For any intervals that do not overlap, in words, what does this imply about the significance of the differences in the true proportions? Intervals that do not overlap do not give any indication about the difference of the true population proportion The intervals for black and white do not overlap, which means that there is no significant difference in their population proportions. The intervals for black and white do not overlap, which means that there is a significant difference in their population proportions. There are no intervals that do not overlap, implying that there is not a significant difference in their population proportions. 7. Six different national brands of chocolate chip cookies were randomly selected at the supermarket. The grams of fat per serving are as follows: 9; 9; 11; 7; 8; 8. Assume the underlying distribution is approximately normal. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) Calculate a 90% confidence interval for the population average grams of fat per serving of chocolate chip cookies sold in supermarkets. (i) State the confidence interval. (Round your answers to two decimal places.) ( __ , __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to two decimal places.) Part (b) If you wanted a smaller error bound while keeping the same level of confidence, what should have been changed in the study before it was done? decrease the sample size determine the population standard deviation nothing can be changed to guarantee a smaller error bound increase the sample size 8. A sample of 15 small bags of the same brand of candies was selected. Assume that the population distribution of bag weights is normal. The weight of each bag was then recorded. The mean weight was 2 ounces with a standard deviation of 0.14 ounces. The population standard deviation is known to be 0.1 ounce. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) (i) x = (ii) = (iii) sx = (iv) n= (v) n1= Part (b) Define the Random Variable X, in words. the weight, in ounces, of a piece of candy the number of candies in a bag the average weight of the candies the weight, in ounces, of one bag of candy Part (c) Define the Random Variable x , in words. the average number of candies for the sample of 15 bags the average weight, in ounces, of the pieces of candy the average weight, in ounces, for the sample of 15 bags the average number of candies from each bag that weigh 2 ounces Part (d) Which distribution should you use for this problem? (Round your answers to three decimal places.) x ~ __ ( __ , __ ) Explain your choice. The Student's t-distribution should be used because the sample size is small. The Student's t-distribution should be used because the sample standard deviation is given. The standard normal distribution should be used because the population standard deviation is known. The standard normal distribution should be used because the sample standard deviation is known. Part (e) Construct a 90% confidence interval for the population average weight of the candies. (i) State the confidence interval. (Round your answers to three decimal places.) ( __ , __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to three decimal places.) Part (f) Construct a 98% confidence interval for the population average weight of the candies. (i) State the confidence interval. (Round your answers to three decimal places.) ( __ , __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to three decimal places.) Part (g) Explain why the confidence interval in part (f) is larger than the confidence interval in part (e). The confidence interval in part (f) is larger than the confidence interval in part (e) because the mean weight changes for each sample. The confidence interval in part (f) is larger than the confidence interval in part (e) because a small sample size is being used. The confidence interval in part (f) is larger than the confidence interval in part (e) because a larger level of confidence increases the error bound, making the interval larger. The confidence interval in part (f) is larger than the confidence interval in part (e) because the population standard deviation changes for each sample. Part (h) Give an interpretation of what the interval in part (f) means. We are 98% confident that a small bag of candies weighs between these values. We are 98% confident that the average weight of the sample of 15 small bags of candies is between these values. There is a 98% chance that a small bag of candies weighs between these values. We are 98% confident that the true population average weight of all small bags of candies is between these values. A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length of time they last is approximately normal. Researchers in a hospital used the drug on a random sample of 9 patients. The effective period of the tranquilizer for each patient (in hours) was as follows: 2.5; 2.9; 3.1; 2.3; 2.3; 2.2; 2.8; 2.1; and 2.4. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) (i) x = (rounded to two decimal places) (ii) sx = (rounded to two decimal places) (iii) n= (iv) n1= Part (b) Define the Random Variable X, in words. the number of patients that were given a tranquilizer time, in minutes, of the effectiveness of a tranquilizer the number of tranquilizers the hospital used for each patient time, in hours, of the effectiveness of a tranquilizer Part (c) Define the Random Variable x , in words. the average number of tranquilizers the pharmaceutical company dispenses to hospitals the average length, in hours, of the effectiveness period of tranquilizers the average number of tranquilizers used for each patient the average length, in minutes, of the effectiveness period of tranquilizers Part (d) Which distribution should you use for this problem? ____ (Enter your answer in the form z or tdf where df is the degrees of freedom.) Explain your choice. The Student's t-distribution should be used because the sample standard deviation is known and the sample size is small. The standard normal distribution should be used because the sample standard deviation is known. The Student's t-distribution should be used because the sample size is small. The standard normal distribution should be used because the population standard deviation is known. Part (e) Construct a 95% confidence interval for the population average length of time. (i) State the confidence interval. (Round your answers to two decimal places.) ( __ , __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to two decimal places.) Part (f) What does it mean to be "95% confident" in this problem? This means that we are 95% confident that the average length of effectiveness of tranquilizers in the sample of 9 people is between the interval values. We are 95% confident that the effectiveness of a tranquilizer lies between the interval values. This means that the chances of a tranquilizer being effective is 95%. This means that if intervals are created from repeated samples, 95% of them will contain the true population average length of effectiveness of tranquilizers. A university conducted a study of whether running is healthy for men and women over age 50. During the first eight years of the study, 1.6% of the 453 members of a fitness association died. We are interested in the proportion of people over 50 who ran and died in the same eight-year period. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) Define the Random Variables X and P', in words. X is the proportion of all runners that die, and P' is the number of all runners that die. X is the proportion of people who ran and died in the eight-year period, and P' is the number of people who ran and died in the eight-year period. X is the number of people who ran and died in the eight-year period, and P' is the proportion of people who ran and died in the eight-year period. X is the number of all runners that die, and P' is the proportion of all runners that die. Part (b) Which distribution should you use for this problem? (Round your answers to four decimal places.) P' ~ ( __ , __ ) Explain your choice. The Student's t-distribution should be used because we do not know the standard deviation. The binomial distribution should be used because the two outcomes are "the runner died" and "the runner did not die." The Student's t-distribution should be used np q because 10, which implies a small sample. The standard normal distribution should be used because we are interested in proportions. Part (c) Construct a 97% confidence interval for the population proportion of people over 50 who ran and died in the same eight-year period. (i) State the confidence interval. (Round your answers to four decimal places.) ( __ , __ ) (ii) Sketch the graph. (iii) Calculate the error bound. (Round your answer to four decimal places.) Part (d) Explain what a "97% confidence interval" means for this study. There is a 97% chance that a runner over the age of 50 will die. We are 97% confident that fewer than 3% of people over the age of 50 are runners. We are 97% confident that the number of deaths of runners over the age of 50 is less than 3 people. We are 97% confident that the population proportion of runners over the age of 50 that will die in the next eight years is between the interval values