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
Since your sample size is small, you can use a one-sample t-test formula.
With your data:
z = (2.8 - 3)/(0.3/√10)
Finish the calculation.
Next, check a t-table using 9 degrees of freedom (which is n - 1) for a one-tailed test at .05 level of significance. Compare your z-statistic above to the critical value in the table to determine the outcome.
Substitute t for z in my statements. Sorry for any confusion.
To determine if there is enough evidence to indicate that the average GPA of the group is less than 3.0, we need to perform a hypothesis test. Here's how you can do it:
1. State the null hypothesis (H0) and alternative hypothesis (Ha):
- Null hypothesis: The average GPA of the group is equal to or greater than 3.0.
- Alternative hypothesis: The average GPA of the group is less than 3.0.
2. Set the significance level (α):
- In this case, the significance level is 5%, which means we are willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is actually true).
3. Calculate the test statistic:
- The test statistic for this situation is the z-score, which indicates how many standard deviations the sample mean is away from the population mean.
- The formula for calculating the z-score is: z = (x̄ - μ) / (σ / sqrt(n)), where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
- In this case, x̄ = 2.8 (the average GPA of the group), μ = 3.0 (the hypothesized population mean), σ = 0.3 (the population standard deviation), and n = 10 (the sample size).
4. Find the critical value:
- Since our alternative hypothesis is that the average GPA is less than 3.0, we are conducting a one-tailed test using the lower critical value.
- To find the critical value, we need to look up the z-score corresponding to a 5% significance level (α = 0.05) in the left-tail of the standard normal distribution table.
5. Compare the test statistic with the critical value:
- If the test statistic is less than the critical value, we reject the null hypothesis.
- If the test statistic is greater than or equal to the critical value, we fail to reject the null hypothesis.
6. Make a decision and interpret the results:
- If the null hypothesis is rejected, it means there is enough evidence to indicate that the average GPA of the group is less than 3.0.
- If the null hypothesis is not rejected, it means there is not enough evidence to indicate that the average GPA of the group is less than 3.0.
By following these steps and computing the test statistic, you can determine whether the group of 10 students presents enough evidence to indicate that the average GPA is less than 3.0 at a 5% significance level.