Question

Your research supervisor wants you to test the null hypothesis H₀: μ = 25 against the one-sided alternative hypothesis H_a: μ < 25. The population has a normal distribution with a variance of 16. You are told to use a sample size of 100 and a rejection region of  .

State the probability of a Type II error for this test of significance to four digits to the right of the decimal point under the alternative hypothesis that  μ = 24.

Answer 1

Ho = 25

n = 100

$\\s = 4 \\\bar{X} = 24 \\\alpha = 0.05$

The test hypothesis is

$\\Null\;Hypothesis --> H_0: \mu = \mu_0 \\Alternate\;Hypothesis --> H_1: \mu < \mu_0$

Now, the value of test static can be found out by following formula:

$\\t_0 = \frac{\bar{X} - \mu_0}{s/\sqrt{n}} \\t_0 = \frac{24.0- 25.0}{4.0/\sqrt{100}} \\t_0 = -2.5$

Since the sample size is n = 100, degrees of freedom on the t-test statistic are n-1 = 100-1 = 99

This implies that

$t_{\alpha, n-1} = t_{0.05, 99} = 1.66$

Since, the t distribution is symmetric about zero, so $-t_{0.05,99}$

Since , we reject the null hypothesis $H_0:\mu=25.0$ in favor of the alternative hypothesis $H_1:\mu >25.0 \;at\; \alpha = 0.05.$