Consider the following null and alternative hypotheses:
$$
H_{0}: \mu=120 \text { versus } H_{1}: \mu>120
$$
A random sample of 81 observations taken from this population produced a sample mean of $123.5 .$ The population standard deviation is known to be 15 .
a. If this test is made at the $2.5 \%$ significance level, would you reject the null hypothesis? Use the critical-value approach.
b. What is the probability of making a Type I error in part a?
c. Calculate the $p$ -value for the test. Based on this $p$ -value, would you reject the null hypothesis if $\alpha=.01 ?$ What if $\alpha=.05$ ?