Test Hypotheses about a Population Proportion
-It is desired to test $\mathrm { H } _ { 0 } : \mu = 8$ against $\mathrm { H } _ { 1 } : \mu \neq 8$ using $\alpha = 0.05$. The population in question is uniformly distributed with a standard deviation of 1.0. A random sample of 100 will be drawn from this population. If $\mu$ is really equal to $7.9$, what is the value of $\beta$ associated with this test?
A) $0.1700$
B) $0.8300$
C) $0.3300$
D) $0.0850$

Question

Quizplus · Accepted Answer

To calculate $\beta$ (the probability of a Type II error), we first need to determine the z-scores corresponding to the critical values for a two-tailed test with $\alpha = 0.05$. Since the population is uniformly distributed but we are using a large sample size (n=100), we can apply the Central Limit Theorem and assume the sampling distribution of the mean is approximately normal.Given $\alpha = 0.05$, the critical z-values are $\pm 1.96$ (for a two-tailed test). The standard error (SE) of the mean is $\sigma / \sqrt{n} = 1 / \sqrt{100} = 0.1$.To find $\beta$, we need to calculate the z-score for the actual mean ($\mu = 7.9$) under the null hypothesis ($\mu_0 = 8$):$$z = \frac{\mu - \mu_0}{SE} = \frac{7.9 - 8}{0.1} = -1.0$$This z-score falls within the non-rejection region (between $-1.96$ and $1.96$). To find $\beta$, we look at the probability of failing to reject $H_0$ when $\mu = 7.9$, which involves finding the area under the normal curve to the right of $z = -1.0$ and to the left of $z = 1.0$ (since it's a two-tailed test and we're interested in the probability of being within the acceptance region).The z-score of $-1.0$ corresponds to a left-tail probability of approximately $0.1587$. Since the test is two-tailed, we double this value (for both tails) to get the probability of not rejecting $H_0$ when it is false ($\beta$):$\beta \approx 2 \times 0.1587 = 0.3174$However, none of the provided options exactly match this calculation. The closest option given is $0.3300$, but based on the standard approach to calculating $\beta$ for a two-tailed test with the given parameters, there seems to be a misunderstanding or misinterpretation in the calculation or the options provided. Given the discrepancy, the correct approach to finding $\beta$ involves calculating the probability of not rejecting $H_0$ when it is false, using the z-score for the actual mean under the null hypothesis. However, without the exact probabilities corresponding to the specific z-scores in the options, and given the calculation error in the explanation, the intended correct answer based on standard statistical methods and the provided information cannot be accurately determined from the options given. Revisiting the calculation and considering the standard approach to such problems, the correct answer should involve calculating $\beta$ based on the z-score for the actual mean under the null hypothesis and the standard error of the mean. However, the explanation provided does not match the options given, indicating a potential error in the calculation or in the interpretation of the options.

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

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