\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}

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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.
**Summary:** - **Test Type:** Left-tailed - **Test Statistic (z):** -4.47 - **P-value:** ≈ 0.0000035 - **Null Hypothesis (H₀):** p = 0.10 - **Conclusion:** Reject H₀; there is sufficient evidence to support that fewer than 10% of treated subjects experience headaches.

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