For Problem 10.3, suppose that the population standard deviation is not known, which typically would be the case, and the population of income figures is not assumed to be normally distributed. For a sample of n = 30 households, the sample standard deviation is s = $2,000 and the sample mean remains X = $44,000. Test the null hypothesis that the mean household income in the population is at least $45,000, using the 5 percent level of significance.
Problem 10.3
With reference to Problems 10.1 and 10.2, the prospective developer is not really concerned about the possibility that the average household income is higher than the claimed $45,000, but only that it might be lower. Accordingly, reformulate the null and alternate hypotheses and carry out the appropriate statistical test, still giving the benefit of the doubt to the community representative’s claim.
Problem 10.1
A representative of a community group informs the prospective developer of a shopping center that the average income per household in the area is $45,000. Suppose that for the type of area involved household income can be assumed to be approximately normally distributed, and that the standard deviation can be accepted as being equal to σ = $2,000, based on an earlier study. For a random sample of n = 15 households, the mean household income is found to be X = $44,000. Test the null hypothesis that μ = $45,000 by establishing critical limits of the sample mean in terms of dollars, using the 5 percent level of significance.
Problem 10.2
Test the hypothesis in Problem 10.1, by using the standard normal variable z as the test statistic.