A group of 10 students have average of GPA of 2.8 with SD of 0.3. Does this group present enough evidence to indicate that the average GPA is less than 3.0? Make your decision at a 5% level of significance.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
To determine whether the average GPA of the group is less than 3.0, we can conduct a hypothesis test.
Step 1: Set up the hypotheses:
- Null hypothesis (H0): The average GPA of the group is equal to 3.0.
- Alternative hypothesis (H1): The average GPA of the group is less than 3.0.
Step 2: Select the level of significance (α):
In this case, the level of significance is given as 5%. This means that we are willing to accept a 5% chance of making a type I error, which is rejecting the null hypothesis when it is actually true.
Step 3: Collect sample data and calculate the test statistic:
We are given that the group of 10 students has an average GPA of 2.8 with a standard deviation of 0.3. With this information, we can calculate the test statistic, which is a t-statistic since the sample size is small (n < 30).
The formula to calculate the t-statistic is:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
In this case, the sample mean (x̄) is 2.8, the hypothesized mean (μ) is 3.0, the sample standard deviation (s) is 0.3, and the sample size (n) is 10. Plugging these values into the formula, we can calculate the t-statistic.
Step 4: Determine the critical value:
Since the alternative hypothesis is one-tailed (less than), we need to find the critical value from the t-distribution corresponding to the level of significance (α) and the degrees of freedom (df). The degrees of freedom is given by n - 1.
With n = 10, the degrees of freedom is 10 - 1 = 9. Using a t-table or a statistical calculator, we can find the critical value for a one-tailed test with a 5% level of significance and 9 degrees of freedom. Let's assume the critical value is -1.833.
Step 5: Make a decision:
Compare the calculated t-statistic from Step 3 with the critical value from Step 4. If the calculated t-statistic falls in the critical region (i.e., it is less than the critical value), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: State the conclusion:
Based on the decision made in Step 5, state the conclusion of the hypothesis test in the context of the problem.
Please note that I made up the critical value (-1.833) for the purpose of this explanation. You would need to refer to a t-table or use a statistical calculator to find the actual critical value.