A university interested in tracking its honors program believes that the proportion of graduates with a GPA of 3.00 or below is less than 0.11. In a sample of 280 graduates, 25 students have a GPA of 3.00 or below. The value of the test statistic and its associated p-value are __________.Multiple Choicet279 = −1.11 and p-value = 0.0768z = −1.11 and p-value = 0.0768t279 = −1.11 and p-value = 0.1335z = −1.11 and p-value = 0.1335
Question
A university interested in tracking its honors program believes that the proportion of graduates with a GPA of 3.00 or below is less than 0.11. In a sample of 280 graduates, 25 students have a GPA of 3.00 or below. The value of the test statistic and its associated p-value are __________.Multiple Choicet279 = −1.11 and p-value = 0.0768z = −1.11 and p-value = 0.0768t279 = −1.11 and p-value = 0.1335z = −1.11 and p-value = 0.1335
Solution
To answer this question, we first need to identify the type of test statistic we should use. Since we are dealing with proportions, we should use a z-test, not a t-test.
Next, we calculate the test statistic. The formula for the z-score is:
z = (p̂ - p₀) / √(p₀(1 - p₀) / n)
where:
- p̂ is the sample proportion
- p₀ is the population proportion
- n is the sample size
In this case:
- p̂ = 25 / 280 = 0.0893
- p₀ = 0.11
- n = 280
Substituting these values into the formula gives:
z = (0.0893 - 0.11) / √(0.11(1 - 0.11) / 280) = -1.11
The p-value is the probability of observing a result as extreme as the test statistic, assuming the null hypothesis is true. In this case, the null hypothesis is that the proportion of graduates with a GPA of 3.00 or below is 0.11.
We can find the p-value using a z-table or a statistical software. The p-value associated with a z-score of -1.11 is approximately 0.1335.
So, the correct answer is "z = −1.11 and p-value = 0.1335".
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