A cellular phone company advertises that the average age of its customers is lower than 30 years. It is known that the customer ages follow a normal distribution. A random sample of 16 of this company's customers responded as follows, when asked about their age: 35, 19, 25, 33, 32, 51, 65, 43, 44, 29, 33, 21, 34, 30, 25, 57.
a) Calculate the sample mean and sample variance.
b) Develop an appropriate hypothesis to test the company's claim at the 5% level of significance.
c) If the same sample results had been obtained from a random sample of 25 customers, could the company's claim be rejected at a lower level of significance than in part (b)?
d) Suppose that the alternative hypothesis has been one-sided and that it was set as Ha : μ < 30. Make a graph to visualize the problem and state, without doing the calculations, whether the p-value of the test (the level of significance needed to reject the null hypothesis) would be higher than, lower than, or the same as found in (c).
e) Using the original data, test at the 5% significance level that the population standard deviation of customer ages is greater than 15.
f) Using the original data, estimate the population standard deviation with 90% probability.