Test Hypotheses about a Population Proportion
-It is desired to test $\mathrm { H } _ { 0 } : \mu = 45$ against $\mathrm { H } _ { 1 } : \mu

Question

Test Hypotheses about a Population Proportion
-It is desired to test $\mathrm { H } _ { 0 } : \mu = 45$ against $\mathrm { H } _ { 1 } : \mu < 45$ 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 40 , what is the probability that the hypothesis test would lead the investigator to commit a Type II error?
A) $0.1469$
B) $0.8531$
C) $0.3531$
D) $0.2938$

Quizplus · Accepted Answer

The probability of committing a Type II error ($\beta$) is found by calculating the probability that the test statistic falls in the non-rejection region when the alternative hypothesis is true. Given $\mu_0 = 45$, $\mu = 40$, $\sigma = 15$, $n = 49$, and $\alpha = 0.10$, we first find the z-score for the critical value corresponding to $\alpha = 0.10$ in a left-tailed test, which is approximately -1.28. Then, we calculate the z-score for $\mu = 40$ using the formula $z = (\bar{x} - \mu_0) / (\sigma / \sqrt{n})$. Plugging in the values, we get $z = (40 - 45) / (15 / \sqrt{49}) = -2.333$. The probability of a Type II error is the probability of observing a z-score greater than -1.28 when the true z-score is -2.333, which corresponds to the cumulative probability up to -1.28 minus the cumulative probability up to -2.333. Using standard normal distribution tables or a calculator, we find this probability to be approximately 0.1469.

Test Hypotheses About a Population Proportion -It Is Desired to Test

Test Hypotheses About a Population Proportion
-It Is Desired to Test