Chapter 10 - Section 10.6 - Assess Your Understanding - Applying the Concepts - Page 521: 17

Power curve for the power of test is obtained as follows:

The sample proportion is distributed normally with average \[{{\mu }_{{\hat{p}}}}=0.30\] and standard deviation \[\begin{align} & {{\sigma }_{{\hat{p}}}}=\sqrt{\frac{{{p}_{0}}\left( 1-{{p}_{0}} \right)}{n}} \\ & =\sqrt{\frac{0.30\left( 1-0.30 \right)}{300}} \\ & =0.02646 \end{align}\] Any statistics that is below the critical value \[{{z}_{0.05}}=-1.645\] will lead to reject the null hypothesis. So, reject\[{{H}_{0}}\] if $\hat{p}(0.257)=P(z>\frac{0.257-0.2}{\sqrt{\frac{0.2(1-0.2)}{300}}})$ \[\begin{align} & =P\left( z>\frac{0.057}{0.02309} \right) \\ & =P\left( z>2.47 \right) \\ & =0.0068 \end{align}\] The value of \[\beta \] is 0.0068. Probability of \[{{H}_{0}}\] rejection when \[{{H}_{1}}\] is true is\[1-\beta \]. The value \[1-\beta \] is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows: \[\begin{align} & 1-\beta =1-0.0068 \\ & =0.9932 \end{align}\] Hence, power of test is 0.9932. Section 2: If the correct population proportion is 0.22, then average of the sampling distribution \[{{\mu }_{{\hat{p}}}}=0.22\] and standard deviation is \[\begin{align} & {{\sigma }_{{\hat{p}}}}=\sqrt{\frac{{{p}_{0}}\left( 1-{{p}_{0}} \right)}{n}} \\ & =\sqrt{\frac{0.22\left( 1-0.22 \right)}{300}} \\ & =0.02398 \end{align}\] Now, calculate the probability of Type II error: \[\begin{align} & \beta =P\left( \text{Type II error} \right) \\ & =P\left( \text{not rejecting }{{H}_{0}}\text{ when }{{H}_{1}}\text{ is true} \right) \\ & =P\left( \hat{p}>0.257 \right) \\ & =P\left( z>\frac{0.257-0.22}{\sqrt{\frac{0.22\left( 1-0.22 \right)}{300}}} \right) \end{align}\] \[\begin{align} & =P\left( z>\frac{0.037}{0.02398} \right) \\ & =P\left( z>1.55 \right) \\ & =0.0606 \end{align}\] The value of \[\beta \] is 0.0606. Probability of \[{{H}_{0}}\] rejection when \[{{H}_{1}}\] is true is\[1-\beta \]. The value \[1-\beta \] is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows: \[\begin{align} & 1-\beta =1-0.0606 \\ & =0.9394 \end{align}\] Hence, power of test is 0.9394. Section 3: If the accurate population proportion is 0.24, then average of the sampling distribution \[{{\mu }_{{\hat{p}}}}=0.24\] and standard deviation is \[\begin{align} & {{\sigma }_{{\hat{p}}}}=\sqrt{\frac{{{p}_{0}}\left( 1-{{p}_{0}} \right)}{n}} \\ & =\sqrt{\frac{0.24\left( 1-0.24 \right)}{300}} \\ & =0.02466 \end{align}\] Now, calculate the probability of Type II error: \[\begin{align} & \beta =P\left( \text{Type II error} \right) \\ & =P\left( \text{not rejecting }{{H}_{0}}\text{ when }{{H}_{1}}\text{ is true} \right) \\ & =P\left( \hat{p}>0.257 \right) \\ & =P\left( z>\frac{0.257-0.24}{\sqrt{\frac{0.24\left( 1-0.24 \right)}{300}}} \right) \end{align}\] \[\begin{align} & =P\left( z>\frac{0.017}{0.02466} \right) \\ & =P\left( z>0.69 \right) \\ & =0.2451 \end{align}\] The value of \[\beta \] is 0.2451. Probability of \[{{H}_{0}}\] rejection when \[{{H}_{1}}\] is true is\[1-\beta \]. The value \[1-\beta \] is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows: \[\begin{align} & 1-\beta =1-0.2451 \\ & =0.7549 \end{align}\] Hence, the power of test is 0.7549. Section 4: If the accurate population proportion is 0.26, then average of the sampling distribution \[{{\mu }_{{\hat{p}}}}=0.26\] and standard deviation is \[\begin{align} & {{\sigma }_{{\hat{p}}}}=\sqrt{\frac{{{p}_{0}}\left( 1-{{p}_{0}} \right)}{n}} \\ & =\sqrt{\frac{0.26\left( 1-0.26 \right)}{300}} \\ & =0.02532 \end{align}\] Now, calculate the probability of Type II error: \[\begin{align} & \beta =P\left( \text{Type II error} \right) \\ & =P\left( \text{not rejecting }{{H}_{0}}\text{ when }{{H}_{1}}\text{ is true} \right) \\ & =P\left( \hat{p}>0.257 \right) \\ & =P\left( z>\frac{0.257-0.26}{\sqrt{\frac{0.26\left( 1-0.26 \right)}{300}}} \right) \end{align}\] \[\begin{align} & =P\left( z>\frac{-0.03}{0.02532} \right) \\ & =P\left( z>-0.12 \right) \\ & =0.5478 \end{align}\] The value of \[\beta \] is 0.5478. Probability of \[{{H}_{0}}\] rejection when \[{{H}_{1}}\] is true is\[1-\beta \]. The value \[1-\beta \] is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows: \[\begin{align} & 1-\beta =1-0.5478 \\ & =0.4522 \end{align}\] Hence, the power of test is 0.4522. Section 5: If the accurate population proportion is 0.28, then average of the sampling distribution \[{{\mu }_{{\hat{p}}}}=0.28\] and standard deviation is \[\begin{align} & {{\sigma }_{{\hat{p}}}}=\sqrt{\frac{{{p}_{0}}\left( 1-{{p}_{0}} \right)}{n}} \\ & =\sqrt{\frac{0.28\left( 1-0.28 \right)}{300}} \\ & =0.02592 \end{align}\] Now, calculate the probability of Type II error: \[\begin{align} & \beta =P\left( \text{Type II error} \right) \\ & =P\left( \text{not rejecting }{{H}_{0}}\text{ when }{{H}_{1}}\text{ is true} \right) \\ & =P\left( \hat{p}>0.257 \right) \\ & =P\left( z>\frac{0.257-0.28}{\sqrt{\frac{0.28\left( 1-0.28 \right)}{300}}} \right) \end{align}\] \[\begin{align} & =P\left( z>\frac{-0.023}{0.02592} \right) \\ & =P\left( z>-0.89 \right) \\ & =0.8133 \end{align}\] The value of \[\beta \] is 0.8133. Probability of \[{{H}_{0}}\] rejection when \[{{H}_{1}}\] is true is\[1-\beta \]. The value \[1-\beta \] is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows: \[\begin{align} & 1-\beta =1-0.8133 \\ & =0.1867 \end{align}\] Hence, the power of test is 0.1867. Section 6: If the accurate population proportion is 0.29, then average of the sampling distribution \[{{\mu }_{{\hat{p}}}}=0.29\] and standard deviation is \[\begin{align} & {{\sigma }_{{\hat{p}}}}=\sqrt{\frac{{{p}_{0}}\left( 1-{{p}_{0}} \right)}{n}} \\ & =\sqrt{\frac{0.29\left( 1-0.29 \right)}{300}} \\ & =0.02620 \end{align}\] Now, the probability of Type II error is calculated as follows: \[\begin{align} & \beta =P\left( \text{Type II error} \right) \\ & =P\left( \text{not rejecting }{{H}_{0}}\text{ when }{{H}_{1}}\text{ is true} \right) \\ & =P\left( \hat{p}>0.257 \right) \\ & =P\left( z>\frac{0.257-0.29}{\sqrt{\frac{0.29\left( 1-0.29 \right)}{300}}} \right) \end{align}\] \[\begin{align} & =P\left( z>\frac{-0.023}{0.02620} \right) \\ & =P\left( z>-1.26 \right) \\ & =0.8962 \end{align}\] The value of \[\beta \] is 0.8962. Probability of \[{{H}_{0}}\] rejection when \[{{H}_{1}}\] is true is\[1-\beta \]. The value \[1-\beta \] is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows: \[\begin{align} & 1-\beta =1-0.8962 \\ & =0.1038 \end{align}\] Hence, the power of test is 0.1038.