You conduct a survey of a sample of 25 members of this year�s graduating marketing students and find that the average GPA is 3.2. The standard deviation of the sample is 0.4. Over the last 10 years, the average GPA has been 3.0. Is the GPA of this year�s students significantly different from the long-run average? At what alpha level would it be significant?
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to your Z score.
To determine if the GPA of this year's students is significantly different from the long-run average, we can perform a hypothesis test.
First, let's define the null and alternative hypotheses:
- Null Hypothesis (H₀): The GPA of this year's students is not significantly different from the long-run average (μ = 3.0).
- Alternative Hypothesis (H₁): The GPA of this year's students is significantly different from the long-run average (μ ≠ 3.0).
Next, we can calculate the test statistic using the formula:
t = (x̄ - μ) / (s / √n)
Where:
- x̄ is the sample mean (3.2)
- μ is the population mean or the long-run average (3.0)
- s is the sample standard deviation (0.4)
- n is the sample size (25)
Calculating the test statistic:
t = (3.2 - 3.0) / (0.4 / √25)
= 0.2 / (0.4 / 5)
= 0.2 / 0.08
= 2.5
The test statistic is 2.5.
Next, we need to determine the significance level (alpha level) at which we want to test our hypothesis. The alpha level is the probability of rejecting the null hypothesis when it is true. Commonly used alpha levels are 0.05 (5%) and 0.01 (1%).
Let's assume we choose an alpha level of 0.05.
The next step is to compare the test statistic with the critical value(s) from the t-distribution at the chosen alpha level. Since we have a two-tailed test (H₁: μ ≠ 3.0), we need to split the alpha level in half (0.025 for each tail).
Looking up the critical value from the t-distribution table (with 24 degrees of freedom and an alpha level of 0.025), we find the critical value is approximately ±2.064.
Since our test statistic (t = 2.5) is greater than the critical value (±2.064), we can reject the null hypothesis.
Conclusion:
The GPA of this year's students is significantly different from the long-run average at the 0.05 alpha level.
Note: If we had chosen a different alpha level (e.g., 0.01), we would compare the test statistic with the corresponding critical value at that alpha level.