A survey by Men’s Health magazine stated that 40% of men said they used alcohol to reduce
stress. At 0.10, test the claim that a random sample of 100 men was selected and 30 said
that they used alcohol to reduce stress. Use the P-value method.
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To test the claim using the P-value method, we need to follow these steps:
Step 1: State the null and alternative hypotheses.
The null hypothesis (H0) assumes that there is no difference between the observed proportion and the claimed proportion. In this case, the null hypothesis is:
H0: p = 0.40
The alternative hypothesis (Ha) assumes that there is a difference between the observed proportion and the claimed proportion. In this case, the alternative hypothesis is:
Ha: p ≠ 0.40
Step 2: Determine the test statistic.
For this problem, we will use the z-test statistic because the sample size is large (n = 100) and we know the claimed proportion (p = 0.40). The formula for the z-test statistic is:
z = (p̂ - p) / √(p * (1 - p) / n)
where p̂ is the observed proportion, p is the claimed proportion, and n is the sample size.
Step 3: Calculate the test statistic.
Using the given information, we have:
p̂ = 30 / 100 = 0.30
p = 0.40
n = 100
Plugging these values into the formula, we can calculate the test statistic:
z = (0.30 - 0.40) / √(0.40 * (1 - 0.40) / 100)
z = -1 / √(0.40 * 0.60 / 100)
z ≈ -1 / 0.0490
z ≈ -20.41
Step 4: Calculate the P-value.
The P-value is the probability of obtaining a test statistic as extreme as the one calculated under the null hypothesis. Since the alternative hypothesis is two-sided (p ≠ 0.40), we need to calculate the probability in both tails of the standard normal distribution.
Using a standard normal distribution table or a statistical calculator, we can find the P-value associated with z = -20.41. The P-value is very close to 0 (approximately 0.000000000000003), which means it is extremely unlikely to obtain a sample percentage as extreme as the observed one under the null hypothesis.
Step 5: Make a decision.
Finally, we compare the P-value with the significance level (α) to make a decision. In this case, the given significance level α = 0.10.
Since the P-value (approximately 0.000000000000003) is less than the significance level (α = 0.10), we reject the null hypothesis. This means there is sufficient evidence to suggest that the claimed proportion of 40% is not accurate, and the sample data supports the alternative hypothesis that the proportion is different from 40%.
Therefore, the claim that a random sample of 100 men was selected and 30 said they used alcohol to reduce stress can be rejected based on the P-value method with a significance level of 0.10.