Inferential Statistics: Estimation and 13 A new type of speed trap camera is being evaluated to be used for detecting overspeeding vehicles. The current type of camera has the probability of p =0.8 of a successful detection. From a sample of 40 detections using the new cameras, 34 are found successful. (a) Establish a 95% confidence interval for p. (b) Can we conclude that the new type of camera is better? 14 How large a sample is needed in Question 12, If we wish to be 965 confident that our sample proportion will be within 0.02 of the true fraction of the voting population? 15 How large a sample is needed in Question 14, if we wish to be 99% confident that our sample proportion will be within 0.05 of the true proportion of homes in the district that do not pay rent? 16 A researcher is interested in a comparative study of stress among older executives and young executives. From a random sample of 500 older executives, 350 are suffering from stress; whereas from a sample of 450 young executives, 300 are having stress problems. Construct a 90% confidence interval for the difference between the proportion of older and young executives who are suffering from stress. 17 Ten faculties of engineering at different universities were surveyed. The sample contained 250 electrical engineering students, 80 being females and 175 civil engineering students. 40 being females. Compute a 90% confidence interval for the difference between the proportions of female students in these two fields of engineering. Is there a significant difference between the two proportions? 18 A car battery manufacturer claims that his batteries will last. on average. 3 years with a variance of 1 year. If 5 of these batteries have lifespans of 1.9, 2.4. 3.0, 3.5 and 4.2 years, construct a 95% confidence interval for o' and decide if the manufacturer's claim that o' = 1 is valid. Assume the population of the lifespans of the batteries to be approximately normally distributed. 19 A random sample of 20 students obtained a mean of x = 68 and a variance of o' = 9 on a test in mathematics. Assuming the scores are normally distributed. construct a 98% confidence interval for 20 Construct a 90% confidence interval for of /G; in Question 10.6 A random sample of 15 schoolchildren was taken and the weights (in kg) are 40, 35, 38, 41, 45, 44, 47, 50, 32, 33, 42, 46, 39, 31 and 48. Assuming the weights are approximately normal, establish a 95% confidence interval for the population mean weight of the schoolchildren. 7 A random sample of 25 bottles of Vitamin C contains, on average, 325.05 mg with a standard deviation of 0.5. Find the 95% confidence interval for the true mean content of this brand of Vitamin C. Assume that the content is normally distributed. 8 Two random samples were taken from two different normal populations. The first sample of size 30 has a mean of 75 and a standard deviation of 5. The second sample of size 40 (taken from the second population) has a mean of 85 with a standard deviation of 7. Construct a 90% confidence interval for # - u 9 Two groups of students are being compared on their final examination marks. A random sample of 12 students from the first group has a mean of 85% with a standard deviation of 4%. From the second group, a random sample of 10 students has a mean of 81% with a standard deviation of 5%. Assuming normal populations with equal variances, determine a 90% confidence interval for the difference between the population means. 10 A transportation company is considering whether to purchase type R or type S tires for its fleet of trucks. To evaluate the difference in these two types of tires, the company has conducted a test using 15 tires of each type. Each tire is tested on traveling distance in a controlled environment until it wears out. The results are as follows: Type R: x, = 36,300 km. s, = 5.000 km Type S: X, = 38,100 km, s, = 6,100 km Assuming normal populations, construct a 99% confidence interval for H - ly 11 Two hundred graduating students are selected and it is found that 1 14 will be awarded the first class honors degrees. Calculate the 96% confidence interval for the proportion of the graduating students who will receive the first class honors degrees. 12 In a random sample of 500 homes in a district, it is found that 125 do not pay rent. Find the 99% confidence interval for the proportion of homes in this district that do not pay rent.Exercise 7.2 1 A company manufactures a new type of mosquito repellents with the period of effectiveness that is approximately normally distributed with a standard deviation of 30 hours. A sample of 40 repellents was taken and the average period of effectiveness is 290 hours. Calculate a 96% confidence interval for the true mean of the period of effectiveness. 2 A random sample of 100 automobile owners shows that an automobile is driven on average, 23,500 kilometers per year with a standard deviation of 3,900 kilometers. Construct a 99% confidence interval for the average number of kilometers an automobile is driven annually. 3 How large a sample is needed in Question 1 if we wish to be 96% confident that our sample mean will be within 10 hours of the true mean? 4 An operations manager wants to evaluate the mean time to assemble a product, and it is known that the population standard deviation is 30 seconds. If he wishes to be 95% confident that the sample mean will be within 10 seconds of the population mean, how large should the sample size be? 5 A machine dispenses sugar into 1 kg plastic bags. A sample of nine bags of sugar was taken and the average weight is 1.0056 kg with a standard deviation of 0.0246. Assuming a normal population, determine a 99% confidence interval for the mean weight