The IQ scores of children in a city are assumed to follow a normal distribution with unknown mean μ and known variance σ2. A random sample IQ scores with a size of 25 is drawn by a researcher. Based on this sample, a 97% confidence interval for μ is found to be (91.49, 104.51).
(a) Show that σ = 15.
(b) Another random sample of IQ scores with a size of 11 drawn and the scores are recorded as follows:
134, 95, 102, 99, 91, 111, 127, 99, 101, 113, 105
Based on the combined information of the two samples, test, at the 0.05 significance levels, whether or not the mean IQ score of the children in the city is less than the national average of 105.