Question 1181349: According to a study conducted, 48.9% of males aged 20 to 39 years consume the recommended daily requirement of calcium. After an aggressive marketing campaign, a separate survey was conducted and they found that 21 of 35 randomly selected males aged 20 to 39 of them consume the recommended daily requirement of calcium. At the α = 0.10 level of significance, is there evidence to conclude that the percentage of males aged 20 to 39 who consume the recommended daily requirement of calcium has increased?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to conduct a hypothesis test to determine if the marketing campaign was effective:
**1. State the Hypotheses:**
* **Null Hypothesis (H₀):** The percentage of males aged 20-39 who consume the recommended daily requirement of calcium has *not* increased. (p ≤ 0.489)
* **Alternative Hypothesis (H₁):** The percentage of males aged 20-39 who consume the recommended daily requirement of calcium *has* increased. (p > 0.489) This is a right-tailed test.
**2. Determine the Level of Significance:**
α = 0.10
**3. Calculate the Sample Proportion (p̂):**
p̂ = (Number who consume recommended calcium) / (Total number surveyed)
p̂ = 21 / 35
p̂ = 0.6
**4. Calculate the Test Statistic (z-score):**
We use a z-test for proportions because the sample size is large enough.
z = (p̂ - p) / sqrt[p(1-p)/n]
z = (0.6 - 0.489) / sqrt[(0.489 * 0.511) / 35]
z = 0.111 / sqrt(0.00714)
z = 0.111 / 0.0845
z ≈ 1.31
**5. Determine the Critical Value (or P-value):**
* **Critical Value Approach:** For a right-tailed test with α = 0.10, the critical z-value is approximately 1.28 (you can find this using a z-table or calculator). If our calculated z-score is greater than 1.28, we reject the null hypothesis.
* **P-value Approach:** The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. Since this is a right-tailed test, we want the area to the *right* of our z-score (1.31) on the standard normal distribution. Using a z-table or calculator, we find that the p-value is approximately 0.095.
**6. Make a Decision:**
* **Critical Value Approach:** Our calculated z-score (1.31) is *greater* than the critical value (1.28). Therefore, we *reject* the null hypothesis.
* **P-value Approach:** Our p-value (0.095) is *less than* our significance level (0.10). Therefore, we *reject* the null hypothesis.
**7. State the Conclusion:**
There *is* sufficient evidence at the α = 0.10 level of significance to conclude that the percentage of males aged 20 to 39 who consume the recommended daily requirement of calcium has increased after the marketing campaign.
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