suppose we went to test the claim that the average GPA of all students in this university is at least 2.8. however , from out class with 25 students, we find the average GPA is only 2.7 with standard deviation 0.35. can we reject the claim with significance level a=0.10?
To determine whether we can reject the claim that the average GPA of all students in the university is at least 2.8, we need to perform a hypothesis test. Here's how you can go about it:
Step 1: State the null and alternative hypotheses:
The null hypothesis (H0) assumes that the average GPA is at least 2.8.
H0: μ >= 2.8
The alternative hypothesis (Ha) assumes that the average GPA is less than 2.8.
Ha: μ < 2.8
Step 2: Determine the significance level (α):
The significance level (α) is given as 0.10, which means we need to have strong evidence to reject the null hypothesis at a 10% level of significance.
Step 3: Calculate the test statistic:
To calculate the test statistic, we need to use the sample mean (x̄), population standard deviation (σ), sample size (n), and known or estimated population mean (μ).
The given information states that the sample mean (x̄) is 2.7 and the standard deviation (σ) is 0.35. The sample size (n) is 25.
The formula to calculate the test statistic (z-score) in this case is:
z = (x̄ - μ) / (σ / √ n)
Substituting the given values into the formula:
z = (2.7 - 2.8) / (0.35 / √25)
z = -0.1 / (0.35 / 5)
z = -0.1 / 0.07
z ≈ -1.43
Step 4: Calculate the p-value:
Now we need to find the p-value associated with the calculated test statistic (z-score). The p-value represents the probability of obtaining a sample mean as extreme as (or more extreme than) the observed value, assuming the null hypothesis is true.
To find the p-value, we can use a standard normal distribution table or a statistical software. The p-value corresponding to a z-score of -1.43 is approximately 0.0764.
Step 5: Compare the p-value with the significance level:
If the p-value is less than or equal to the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the p-value (0.0764) is greater than the significance level (0.10). Therefore, we fail to reject the null hypothesis.
Conclusion:
Based on the data obtained from the class of 25 students, we do not have enough evidence to reject the claim that the average GPA of all students in the university is at least 2.8, at the 10% level of significance.
To determine whether we can reject the claim that the average GPA of all students in the university is at least 2.8, we can use a hypothesis test. Here are the step-by-step instructions to carry out the test:
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): The average GPA of all students in the university is at least 2.8.
- Alternative hypothesis (Ha): The average GPA of all students in the university is less than 2.8.
Step 2: Determine the significance level (α): The given significance level is α = 0.10.
Step 3: Consider the test statistic:
- Test statistic: We will use the t-test statistic, as the population standard deviation is not known.
Step 4: Set up the critical region:
- Since the alternative hypothesis is that the average GPA is less than 2.8, we have a one-tailed test.
- To determine the critical value, we need to find the t-value associated with the given significance level and the degrees of freedom.
- Degrees of freedom: n - 1 = 25 - 1 = 24 (where n is the sample size)
- Look up the critical t-value from the t-distribution table for a one-tailed test with 24 degrees of freedom and α = 0.10. The critical t-value is approximately -1.711.
Step 5: Calculate the test statistic:
- The t-test statistic formula is: t = (sample mean - hypothesized mean) / (sample standard deviation / √n)
- Plugging in the values from the problem, the test statistic is: t = (2.7 - 2.8) / (0.35 / √25) ≈ -0.42857
Step 6: Make a decision:
- If the test statistic falls within the critical region, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
- Since the test statistic (-0.42857) does not fall within the critical region (-1.711), we fail to reject the null hypothesis.
Step 7: Interpret the results:
- We do not have enough evidence to reject the claim that the average GPA of all students in the university is at least 2.8 at the given significance level of 0.10.