9. Test at the 93 percent level of significance the null hypothesis $H_{0}: p=0.546$ versus the alternative hypothesis $H_{1}: p>0.546$, where $p$ is the population proportion, $n=791$ is the sample size, and $x=440$ is the number of observed "successes". Let $Q_{1}=\hat{p}$ be the sample proportion, $Q_{2}$ the $z$-statistic, and $Q_{3}=1$ if we reject the null hypothesis $H_{0}$, and $Q_{3}=0$ otherwise. Let $Q=\ln \left(3+\left|Q_{1}\right|+2\left|Q_{2}\right|+3\left|Q_{3}\right|\right)$. Then $T=5 \sin ^{2}(100 Q)$ satisfies:- (A) $0 \leq T<1$. (B) $1 \leq T<2$. (C) $2 \leq T<3$. (D) $3 \leq T<4$. (E) $4 \leq T \leq 5$.

Step 1 ：First, we calculate the sample proportion \(\hat{p}\), which is the ratio of the number of observed successes to the sample size. In this case, \(\hat{p} = \frac{x}{n} = \frac{440}{791} = 0.5562579013906448\).

Step 2 ：Next, we calculate the z-statistic using the formula for the test of a population proportion. The z-statistic measures how many standard deviations an element is from the population mean. The formula is \(z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} = \frac{0.5562579013906448 - 0.546}{\sqrt{\frac{0.546(1-0.546)}{791}}} = 0.5794587348059118\).

Step 3 ：We then determine the critical value at the 93 percent level of significance, which is 1.475791028179171.

Step 4 ：We compare the z-statistic to the critical value. If the z-statistic is greater than the critical value, we reject the null hypothesis, and \(Q_{3}=1\). Otherwise, \(Q_{3}=0\). In this case, since 0.5794587348059118 is less than 1.475791028179171, we do not reject the null hypothesis, so \(Q_{3}=0\).

Step 5 ：We then calculate \(Q\) using the formula \(Q = \ln (3 + |\hat{p}| + 2|z| + 3|Q_{3}|) = \ln (3 + |0.5562579013906448| + 2|0.5794587348059118| + 3|0|) = 1.5507861096678492\).

Step 6 ：Finally, we calculate \(T\) using the formula \(T = 5 \sin^{2}(100Q) = 5 \sin^{2}(100 \times 1.5507861096678492) = 4.1302394695460976\).

Step 7 ：We determine which range \(T\) falls into. In this case, \(T = 4.1302394695460976\), so \(T\) falls into the range \(4 \leq T \leq 5\).

Step 8 ：\(\boxed{\text{Final Answer: The correct answer is (E) } 4 \leq T \leq 5}\)