46. Identify the value of the TEST STATISTIC used in a hypothesis test of the following claim and sample data: Claim: "The proportion of symptom-free placebo subjects is less than the proportion of symptom-free subjects who received a new supplement." A medical researcher wants to determine if a new supplement is effective at reducing the duration of the common cold. Healthy subjects are intentionally infected with a cold-causing virus, and then given either a placebo or the new supplement one day later. Of the 267 subjects who were given a placebo, $33 \%$ were symptom-free a week later. And of the 289 subjects who were given the supplement, $56 \%$ were symptom-free after a week. Test the claim at the 0.01 significance level.

Step 1 ：Given that the number of subjects who were given a placebo (n1) is 267 and the proportion of symptom-free placebo subjects (p1) is 0.33.

Step 2 ：Also, the number of subjects who were given the supplement (n2) is 289 and the proportion of symptom-free subjects who received the supplement (p2) is 0.56.

Step 3 ：We first calculate the pooled sample proportion (p_hat) using the formula: \[\hat{p} = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}\] Substituting the given values, we get \[\hat{p} = \frac{267*0.33 + 289*0.56}{267 + 289} = 0.4495503597122302\]

Step 4 ：Next, we calculate the test statistic (Z) using the formula: \[Z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}\] Substituting the calculated and given values, we get \[Z = \frac{(0.33 - 0.56) - 0}{\sqrt{0.4495503597122302*(1-0.4495503597122302)(\frac{1}{267} + \frac{1}{289})}} = -5.446870141488325\]

Step 5 ：Rounding off the test statistic to two decimal places, we get -5.45.

Step 6 ：Final Answer: The value of the test statistic used in the hypothesis test is \(\boxed{-5.45}\)