Question 1185958
Here's how to conduct a hypothesis test for a proportion:

**1. State the Hypotheses:**

*   **Null Hypothesis (H₀):** The proportion of consumers who prefer Pepsi is 50%. (p = 0.50)
*   **Alternative Hypothesis (H₁):** The proportion of consumers who prefer Pepsi is *not* 50%. (p ≠ 0.50)  This is a two-tailed test.

**2. Determine the Level of Significance:**

α = 0.10 (10%)

**3. Calculate the Sample Proportion (p̂):**

p̂ = (Number who prefer Pepsi) / (Total sample size)
p̂ = 146 / 250 = 0.584

**4. Calculate the Test Statistic (z-score):**

z = (p̂ - p) / sqrt[p(1-p) / n]
z = (0.584 - 0.50) / sqrt[(0.50 * 0.50) / 250]
z = 0.084 / sqrt(0.001)
z = 0.084 / 0.0316
z ≈ 2.66

**5. Determine the Critical Value(s) or P-value:**

*   **Critical Value Approach:** Since this is a two-tailed test with α = 0.10, we need to find the z-scores that correspond to the outer 5% of the distribution (2.5% in each tail).  Using a z-table or calculator, the critical values are approximately ±1.645.

*   **P-value Approach:** The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated (in this case, a z-score of 2.66), assuming the null hypothesis is true. Since it is a two-tailed test, we need to double the area in one tail.
    P-value = 2 * P(z > 2.66) ≈ 2 * 0.0039 = 0.0078

**6. Make a Decision:**

*   **Critical Value Approach:** Since the calculated z-score (2.66) falls *outside* the critical region (-1.645 to +1.645), we *reject* the null hypothesis.

*   **P-value Approach:** Since the p-value (0.0078) is *less than* the level of significance (0.10), we *reject* the null hypothesis.

**7. Conclusion:**

There is sufficient evidence at the 10% level of significance to conclude that the proportion of consumers who prefer Pepsi is different from 50%.