It is desired to test $\mathrm { H } _ { 0 } : \mu = 50$ against $\mathrm { H } _ { \mathrm { A } } : \mu \neq 50$ using $\alpha = 0.10$. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If $\mu$ is really equal to 48 , what is the probability that the hypothesis test would lead the investigator to commit a Type II error?
A) $0.2433$
B) $0.8994$
C) $0.7567$
D) $0.1006$

Question

Quizplus · Accepted Answer

The probability of committing a Type II error ($\beta$) is found by calculating the probability that the test statistic falls within the acceptance region when the true mean ($\mu$) is 48, not 50 as hypothesized under $H_0$. Given $\alpha = 0.10$, the critical z-values for a two-tailed test are approximately $\pm1.645$. The standard error (SE) for the mean is $\sigma / \sqrt{n} = 15 / \sqrt{49} = 15 / 7 = 2.14$. The z-scores corresponding to $\mu = 48$ are calculated using the formula $z = (X - \mu) / SE$, where $X$ is the value corresponding to the critical z-values under the null hypothesis. Since the distribution is symmetric and we're interested in the middle portion that leads to failing to reject $H_0$, we calculate the z-scores for the boundaries and use the standard normal distribution to find the probabilities. The calculation leads to a probability of $0.7567$ for committing a Type II error when the true mean is 48, indicating that there's a 75.67% chance the test will not detect the difference from the hypothesized mean of 50.

It Is Desired to Test H0:μ=50\mathrm { H } _ { 0 } : \mu = 50H0​:μ=50

It Is Desired to Test $\mathrm { H } _ { 0 } : \mu = 50$