p-value test, a critical value test, and a confidence interval test.

Question

A statistic professor at UCSD where statistics classes have 350 students, gave his students a satisfaction survey on self-evaluation of how much they learned in the class. The survey was a rating from 1 to 5, where 5 is the highest rating and 1 is the lowest. The summary of the data is as follows. There was a random sample of n = 50, X = 3.72, and s = 0.57. Use a .05 significance level to test the claim that the population of all students has a mean of 4.0 in the self-evaluation satisfaction of how much they learned n the class. Perform a p-value test, a critical value test, and a confidence interval test. Show your work clearly for ALL three tests.

Answer 1

In hypothesis testing, we always start by writing our null and alternative hypothesis. H0: mu = 4.0 H1: mu =/= 4.0. Let's call 4.0 mu0.

Let's find out test statistic z0 = (x_bar - mu0) / (s / sqrt(n)) = (3.72-4) * sqrt(50) / 0.57 = -3.474

P-value method (denoting the inverse cdf of the standard normal as PHI): p_value = 2 * (1 - PHI(z0)) = 0.000515. Since the p-value is less than our significance level, we reject the null hypothesis that mu=4.0

Critical value method: We compare z0 to z_alpha/2 = z such that P(standard normal > z) = alpha/2 = 0.025.

z_0.025 = 1.645. Since the absolute value of z0 is larger than z_alpha/2, we reject the null hypothesis that mu=4.0.

Confidence interval method: We find the confidence interval mu and say that 4.0 is not in the confidence interval.