Women athletes at the a certain university have a long-term graduation rate of 67%. Over the past several years, a random sample of 39 women athletes at the school showed that 23 eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the university is now less than 67%? Use a 1%level of significance.
What is the value of the sample test statistic? And what is the p-value of the test statistic?
To determine whether the population proportion of women athletes who graduate from the university is now less than 67%, we can perform a hypothesis test using the sample data.
Step 1: Define null and alternative hypotheses:
- Null Hypothesis (H₀): The population proportion of women athletes who graduate is equal to or greater than 67%.
- Alternative Hypothesis (H₁): The population proportion of women athletes who graduate is less than 67%.
Step 2: Set significance level (alpha):
In this case, the significance level is given as 1% (0.01).
Step 3: Calculate the test statistic:
The test statistic for testing proportions is the z-score, which is calculated using the formula:
z = (p̂ - p₀) / √((p₀ * (1 - p₀)) / n)
Where:
p̂ = Sample proportion
p₀ = Null hypothesis proportion
n = Sample size
From the given information:
Sample proportion (p̂) = 23 / 39 ≈ 0.58974 (approximately)
Null hypothesis proportion (p₀) = 0.67
Sample size (n) = 39
Calculating the test statistic:
z = (0.58974 - 0.67) / √((0.67 * (1 - 0.67)) / 39)
z ≈ -0.632
Step 4: Calculate the p-value:
The p-value is the probability of observing a test statistic as extreme as the one calculated (or even more extreme) if the null hypothesis is true.
To find the p-value, we need to use the corresponding z-score from the z-table or a statistical calculator. In this case, since the alternative hypothesis is "less than," we are interested in the left-tail area.
Looking up the left-tail area for -0.632 in a z-table or using a statistical calculator, the approximate p-value is 0.264.
Step 5: Conclusion:
Compare the p-value to the significance level. If the p-value is less than the significance level (0.01), we reject the null hypothesis in favor of the alternative hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the p-value (0.264) is greater than the significance level (0.01). Therefore, we fail to reject the null hypothesis.
To summarize:
- The value of the sample test statistic (z) is approximately -0.632.
- The p-value of the test statistic is approximately 0.264.