Question

Test at the α = 0.05 significance level whether the mean of a random sample of size n = 16 is statistically significantly less than 10 if the distribution from which the sample was taken is normal, ?̅= 8.4 and ? 2 = 10.24. a) What is the appropriate test you can use to test the claim? b) What are the null and alternative hypotheses for this test? c) What is your conclusion? d) Find the confidence interval on the mean at 5% level of significance.  

Answer 1

One tail z-test for single mean

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 10
Alternative Hypothesis, Ha: μ < 10

Test statistic,
z = (xbar - mu)/(sigma/sqrt(n))
z = (8.4 - 10)/(3.2/sqrt(16))
z = -2

P-value Approach
P-value = 0.0228
As P-value < 0.05, reject the null hypothesis.

d)
sample mean, xbar = 8.4
sample standard deviation, σ = 3.2
sample size, n = 16
Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (8.4 - 1.96 * 3.2/sqrt(16) , 8.4 + 1.96 * 3.2/sqrt(16))
CI = (6.83 , 9.97)