Shipments of coffee beans are checked for moisture content. A high moisture content indicates water contamination and will result in the shipment being rejected. Let $\mu$ represent the mean water content (in percent by weight) in a shipment. Sixty moisture measurements will be made on beans chosen at random from the shipment. A test of the hypotheses $H _ { 0 } : \mu = 11$ versus $H _ { 1 } : \mu 11$ will be made at the $\alpha = 0.01$ level of significance. Assume the standard deviation of moisture content is $\sigma$= 5.0. Find the power of the test against the alternative $\mu _ { 1 }$=13.
A) 0.7794
B) 0.2206
C) 0.8828
D) -0.7724

Question

Shipments of coffee beans are checked for moisture content. A high moisture content indicates water contamination and will result in the shipment being rejected. Let $\mu$ represent the mean water content (in percent by weight) in a shipment. Sixty moisture measurements will be made on beans chosen at random from the shipment. A test of the hypotheses $H _ { 0 } : \mu = 11$ versus $H _ { 1 } : \mu > 11$ will be made at the $\alpha = 0.01$ level of significance. Assume the standard deviation of moisture content is $\sigma$= 5.0. Find the power of the test against the alternative $\mu _ { 1 }$=13.
A) 0.7794
B) 0.2206
C) 0.8828
D) -0.7724

Quizplus · Accepted Answer

The power of a test is the probability of correctly rejecting the null hypothesis when it is false. To find the power of the test against the alternative $\mu_1 = 13$, we first calculate the z-score for the significance level $\alpha = 0.01$ for a one-tailed test, and then use this to find the probability of observing a test statistic that leads to rejecting $H_0$ when $\mu = 13$. Given $\sigma = 5.0$ and $n = 60$, the calculation involves standardizing the difference between the alternative mean and the hypothesized mean, and then finding the corresponding probability. The correct calculation leads to a power of approximately 0.7794, indicating a high likelihood of correctly rejecting $H_0$ in favor of $H_1$ when the true mean moisture content is 13%.

Shipments of Coffee Beans Are Checked for Moisture Content μ\muμ Represent the Mean Water Content (In Percent by Weight) in High

Shipments of Coffee Beans Are Checked for Moisture Content $\mu$ Represent the Mean Water Content (In Percent by Weight) in High