STAT 1000 Assignment 5 DUE: March 28th (Wed. Eve. Section), March 29th (T/Th. Sections), March 30th (MWF. Sections) SHOW ALL YOUR WORK 1. Hoping to attract more shoppers, a city builds a new public parking garage downtown. The city plans to pay for the structure through parking fees. The consultant, who advised the city on this project, randomly selected 44 weekdays. Daily fees collected averaged $126. Based on data from other parking structures, the consultant will assume parking fees at this parking garage are Normally distributed with a standard deviation of $15. (a) What is a 95% condence interval for the true mean daily income this parking garage will generate from the parking fees? (b) What is a 90% condence interval for the true mean daily income this parking garage will generate from the parking fees? (c) What is a 74% condence interval for the true mean daily income this parking garage will generate from the parking fees? (d) Provide a proper interpretation of the condence interval you found in (c)? 2. A real estate agent would like to estimate the true mean value of all houses in Winnipeg. Suppose it is known that the standard deviation of house values in Winnipeg is $52,480. (a) What sample size is required to estimate the true mean house value in Winnipeg to within $14,000 with 95% condence. (b) What sample size is required to estimate the true mean house value in Winnipeg to within $14,000 with 99% condence. (c) What sample size is required to estimate the true mean house value in Winnipeg to within $3,500 with 95% condence. (d) Suppose it is known that the city of Saskatoon has one third as many houses as Winnipeg. Also suppose that the standard deviation of house values in Saskatoon is the same as in Winnipeg. What sample size would be required to estimate the true mean house value in Saskatoon to within $14,000 with 95% condence? 3. The breaking strength of yarn used in the manufacture of woven carpet material is normally distributed with = 2.4 psi. A random sample of 16 specimens of yarn from a production run were measured for breaking strength, and based on the mean of the sample, x, a condence interval was found to be (128.7, 131.3). (a) What is the condence level, C, of this interval? (b) If we want to increase our condence, what would happen to the length of the condence interval? 4. It is known that driving can be dicult in regions where winter conditions involve snowcovered roads. For cars equipped with all-season tires traveling at 90 km/hr, the mean stopping time in fresh snow is known to be 215 meters with a standard deviation of 2.5 meters. It is often advocated that automobiles in such areas should be equipped with special tires to compensate for such conditions, especially with respect to stopping distance. A manufacturer of tires made for driving in fresh snow claims that vehicles equipped with their tires have a decreased stopping distance. A study was done using a random sample of 9 snow tires from the manufacturer on a snow-covered test track. The tests resulted in a mean stopping distance of 212.9 meters. (a) Calculate a 90% condence interval for the true mean stopping distance for these tires. (b) Provide an interpretation of the condence interval in (a)? (c) Conduct an appropriate hypothesis test at the 10% level of signicance to test whether these tires result in a decreased stopping distance. Show all steps. (d) Could the interval in (a) have been used to conduct the test in (c)? If no, explain why not. If yes, explain why and what your conclusion would have been and why. 5. A machine at AMT & Co. lls 120-ounce jugs with laundry softener. A quality control inspector wishes to test if the machine needs an adjustment or not, which is needed when the machine either overlls or underlls the jugs. Assume the distribution of the ll volume in these jugs is normal. Under standard circumstances, the mean amount should be 120 ounces with a standard deviation of 1 ounce. A simple random sample of 40 jugs were selected and the mean ll volume was found to be 119.5 ounces. (a) Construct a 94% condence interval for the true mean ll volume for all jugs of softener. (b) Conduct a hypothesis test at the 6% level of signicance to determine whether there is evidence that the true mean ll volume for all jugs of softener diers from 120 ounces. (c) Interpret the P-value from the test in (b). (d) Could the interval in (a) have been used to conduct the test in (b)? If no, explain why not. If yes, explain why, and explain what your conclusion would have been and why. 6. When the NDP formed the provincial government in 1999, the mean wait time for a particular type of surgery was 60 days. Public health ocials take a sample of 30 individuals who have had the surgery in 2011 and record the number of days the patients had to wait prior to having surgery. The data are as follows: 9 61 14 18 23 26 31 37 40 41 41 43 52 54 57 58 65 65 67 69 69 72 74 76 76 77 80 81 87 89 The population standard deviation of wait time for this type of surgery is known to be 20 days. (a) Using JMP, create a histogram for this data set. What is the shape of the distribution of wait times? (b) It is clear from the histogram in (a) that wait times do not follow a normal distribution. Why is it nevertheless appropriate to use inference methods which rely on the assumption of normality? (c) Using JMP, conduct an appropriate hypothesis test at the 1% level of signicance to determine whether the true mean wait time has decreased since the NDP formed government. (You will have to write out the hypothesis and the conclusion - JMP will give you the test statistic and the P -value.) (d) Provide a carefully worded interpretation of the P -value calculated in part (a) which can be understood by someone with little or no knowledge of statistics. (e) Use JMP to nd the 95% condence interval for the true mean wait time for surgery. 7. The lifetimes (in hours) of a sample of 30 AA batteries produced by some company are measured. The data are shown below: 58 92 112 117 99 74 28 77 56 75 82 101 63 100 88 37 60 124 68 112 84 40 129 94 95 67 52 118 93 110 It is known that the standard deviation of all AA batteries produced by this company is 25 hours. The sample mean is calculated to be x = 83.5. (a) We would like to use inference procedures to estimate and test for the true mean lifetime of all AA batteries produced by this company. However, we dont know whether lifetimes follow a normal distribution. Explain why it is nevertheless appropriate to use inference procedures which rely on the assumption of normality. (b) Calculate a 97% condence interval for the true mean lifetime of all AA batteries produced by this company. (c) Provide an interpretation of the interval you constructed in (b). (d) Conduct an appropriate hypothesis test at the 3% level of signicance to determine whether the true mean lifetime of all AA batteries produced by this company diers from 75 hours. Use the P-value approach. (e) Provide an interpretation of the P-value of the test in (d). (f) Suppose you had instead used the critical value approach to conduct the test in (d). What would be the decision rule and the conclusion? (g) Use JM P to verify your results in (d). Include the JM P output in your assignment and highlight the value of the test statistic and the P-value. (h) Could the interval in (b) have been used to conduct the test in (d)? If no, explain why not. If yes, explain why, and explain what your conclusion would have been and why