\begin{tabular}{l} Using Technology. In Exercises 5-8, identify the indicated values or interpret the given \\ display. Use the normal distribution as an approximation to the binomial distribution, as de- \\ scribed in Part 1 of this section. Use a 0.05 significance level and answer the following: \\ a. Is the test two-tailed, left-tailed, or right-tailed? \\ b. What is the test statistic? \\ c. What is the P-value? \\ d. What is the null hypothesis, and what do you conclude about it? \\ e. What is the final conclusion? \\ 5. Adverse Reactions to Drug The drug Lipitor (atorvastatin) is used to treat high choles- \\ terol. In a clinical trial of Lipitor, 47 of 863 treated subjects experienced headaches (based on \\ data from Pfizer). The accompanying TI-83/84 Plus calculator display shows results from a \\ test of the claim that fewer than \( 10 \% \) of treated subjects experience headaches. \\ \hline pr \\ \hline \( \mathbf{z =} \)\end{tabular}

Sure, let's work through each part of Exercise 5 step by step. ### **5. Adverse Reactions to Drug** **Context:** - **Drug:** Lipitor (atorvastatin) used to treat high cholesterol. - **Clinical Trial Data:** 47 out of 863 treated subjects experienced headaches. - **Claim Being Tested:** Fewer than 10% of treated subjects experience headaches. - **Significance Level (α):** 0.05 We'll perform a hypothesis test for a population proportion using the normal approximation to the binomial distribution. --- #### **a. Is the test two-tailed, left-tailed, or right-tailed?** **Answer:** The test is **left-tailed**. **Explanation:** - **Null Hypothesis (H₀):** The proportion of subjects experiencing headaches is equal to 10% (p = 0.10). - **Alternative Hypothesis (H₁):** The proportion of subjects experiencing headaches is less than 10% (p < 0.10). Since we're testing if the proportion is **less than** a certain value, it's a left-tailed test. --- #### **b. What is the test statistic?** **Answer:** \( \mathbf{z = -4.47} \) (rounded to two decimal places) **Calculation Steps:** 1. **Sample Proportion (p̂):** \[ \hat{p} = \frac{47}{863} \approx 0.05437 \] 2. **Standard Error (SE):** \[ SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.10 \times 0.90}{863}} \approx \sqrt{\frac{0.09}{863}} \approx \sqrt{0.0001043} \approx 0.01021 \] 3. **Test Statistic (z):** \[ z = \frac{\hat{p} - p_0}{SE} = \frac{0.05437 - 0.10}{0.01021} \approx \frac{-0.04563}{0.01021} \approx -4.47 \] --- #### **c. What is the P-value?** **Answer:** \( \mathbf{P\text{-value} \approx 0.0000035} \) **Explanation:** - For \( z = -4.47 \), the P-value corresponds to the probability of observing a z-score less than -4.47 in the standard normal distribution. - Using standard normal distribution tables or a calculator: \[ P(z < -4.47) \approx 3.5 \times 10^{-6} \] - This is approximately 0.00035%, which is **much less than** the significance level of 0.05. --- #### **d. What is the null hypothesis, and what do you conclude about it?** **Answer:** - **Null Hypothesis (H₀):** \( p = 0.10 \) - **Conclusion:** Reject H₀. **Explanation:** - **Decision Rule:** If P-value < α, reject H₀. - Since \( 0.0000035 < 0.05 \), we reject the null hypothesis. --- #### **e. What is the final conclusion?** **Answer:** There is **sufficient evidence** at the 0.05 significance level to support the claim that **fewer than 10%** of treated subjects experience headaches. **Explanation:** - Rejecting H₀ indicates that the data supports the alternative hypothesis. - Therefore, we conclude that the proportion of subjects experiencing headaches due to Lipitor is significantly **less than 10%**. --- ### **Summary of Results:** - **a.** Left-tailed test - **b.** \( z = -4.47 \) - **c.** P-value ≈ 0.0000035 - **d.** H₀: \( p = 0.10 \); **Conclusion:** Reject H₀ - **e.** Sufficient evidence to support that fewer than 10% of treated subjects experience headaches.