Question 1185958: A market research consultant hired by the Pepsi-Cola Co, is interested in knowing if the proportion of consumers who favor Pepsi-Cola over Coke Classic is different than 50%. A random sample of 250 consumers from the market under investigation were polled and 464 preferred pepsi in a taste test. Using a 10 Level of significance, test the appropriate hypothesis to help answer his question
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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%.
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