A random sample size of N1 was drawn from a population with

A random sample size of N₁ was drawn from a population with s=5.2 and the scores of a quantitative variable X collected for each subject in the sample. A Z-based 95% confidence interval for the mean based on this sample is (14.26, 16.34).

Assume now that you have a sample of size N₂=100 from the same population and that x-bar remains the same: for what values of m₀ the null hypothesis H₀ = m₀ would be rejected now? (also at the 1% significance level and using a two-sided Z-test and this new sample?) Explain why there are values of m₀ such that H₀ = m₀ is reject with N=N₂ but are not rejected when N= N₁? What can you say about the p-value for the two-sided z-test of the null hypothesis H₀ = 16? Compute exact the p-value.

Solution

Mean = (14.26+16.34)/2 = 15.3

14.26 = 15.3 - (1.96 * 5.2) / sqrt(N1))

=> N1 = 96

a) For,N2

z-score = (m0 - X-bar / (s/sqrt(n))

1.96 = (m0 - 15.3) / (5.2/10)

=>m0 = 16.319 Answer

For,N1

z-score = (m0 - X-bar / (s/sqrt(n))

1.96 = (m0 - 15.3) / (5.2/sqrt(96))

=>m0 = 16.3402

b)

z = (16 - 15.3)/(5.2/10) = 1.346

Thereore , P value = 0.9115 Answer