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
To determine if the GPA of this year's students is significantly different from the long-run average, we can perform a hypothesis test using the standard deviation of the sample and the long-run average GPA.
Here are the steps to perform the hypothesis test:
Step 1: State the null hypothesis (H0) and alternative hypothesis (H1):
- Null hypothesis (H0): This year's students' GPA is not significantly different from the long-run average GPA.
- Alternative hypothesis (H1): This year's students' GPA is significantly different from the long-run average GPA.
Step 2: Choose a significance level (alpha):
- The significance level (alpha) determines the threshold for rejecting the null hypothesis. Typically, a significance level of 0.05 (or 5%) is commonly used.
Step 3: Calculate the test statistic:
- In this case, we can use a t-test since the sample size is small (less than 30) and the population standard deviation is unknown. The formula to calculate the t-test statistic is:
t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
Step 4: Determine the critical value:
- Based on the significance level and the degrees of freedom (n-1), the critical value can be found in the t-distribution table.
Step 5: Compare the test statistic with the critical value:
- If the absolute value of the test statistic is greater than the critical value, then we reject the null hypothesis and conclude that the GPA of this year's students is significantly different from the long-run average GPA.
Please provide the sample size (n) and the sample mean to help proceed with the calculations.
To determine if the GPA of this year's students is significantly different from the long-run average, we would need more information. Specifically, we would need the sample mean GPA for the current year.
Once you have the sample mean GPA for this year, you can perform a hypothesis test using the standard deviation of the sample (0.4) and the long-run average GPA (3.0).
Here are the steps to conduct the hypothesis test:
1. Set up the null and alternative hypotheses:
- Null hypothesis (H₀): The GPA of this year's students is not significantly different from the long-run average.
- Alternative hypothesis (H₁): The GPA of this year's students is significantly different from the long-run average.
2. Choose a significance level (also known as alpha level) to determine the threshold for rejecting the null hypothesis. This is typically denoted by α. Common choices for alpha levels are 0.05 (5%) or 0.01 (1%). The alpha level indicates the probability of rejecting the null hypothesis when it is actually true.
3. Calculate the test statistic:
- For this case, you can use a t-test since we have the sample standard deviation. The test statistic formula is:
t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
4. Determine the critical value or p-value:
- If the test statistic falls outside the critical value region, reject the null hypothesis.
- For a two-tailed test (since we're testing for significant differences in either direction), divide the alpha level by 2 and compare the test statistic against the critical values obtained from the t-distribution table.
- Alternatively, you can calculate the p-value, which is the probability of observing a test statistic as or more extreme than the one obtained, given that the null hypothesis is true. If the p-value is less than the chosen alpha level, reject the null hypothesis.
5. Compare the test statistic to the critical value or p-value:
- If the test statistic is greater than the critical value or the p-value is less than the chosen alpha level, reject the null hypothesis. This means that the GPA of this year's students is significantly different from the long-run average.
- If the test statistic is not greater than the critical value or the p-value is greater than the chosen alpha level, fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the GPA of this year's students is significantly different from the long-run average.
Remember to conduct the hypothesis test using the specific data for this year's students to get the final result.