Question

The Wechsler Adult Intelligence Scale (IQ test) is constructed so that Full Scale IQ scores follow a normal distribution, with a mean of 100, and a standard deviation of 15. The mayor of Smart Town believes the county’s residents are smarter than the national average and wants to use it (the intelligence of the residents) as a marketing tool to bring industries to the area.A researcher is hired to conduct a study to determine whether Smart County’s residents have, on average, higher Full Scale IQs than the population. A random sample of 100 people from Smart County were given the IQ test and were found to have an average Full Scale IQ of 105.Which hypothesis test should be used to determine whether the mean Full Scale IQ score of the Smart County residents is higher than the national average?A.z-test for the population meanB.t-test for the population meanC.z-test for the population proportionD.t-test for the population proportion

Answer 1

A. z-test for the population mean

Explanation:

Here, we are dealing with sampling distribution of mean with a given population standard deviation following the standard normal distribution for large sample size. Hypothesis testing which involves large samples (large samples starts from n = 30 upwards) is based on the assumption that the population from which the sample is drawn has a normal distribution and the sampling distribution of the sample statistic is also normal. Even if the population does not have a normal distribution, the sampling distribution of the sample statistic is assumed to be normal due to the central limit theorem since the sample size is large. In this case, since we are to determine whether the mean Full Scale IQ score of the Smart County residents is higher, then a right-tailed z- test will be used and the hypothesis will be H0: μ = μ0 against H1: μ > μ0 where the decision rule is to reject the null hypothesis if the calculated test statistic is greater than the tabulated z value at a given level of significance.