Let's go through the steps to test the claim that the mean GPA of ASBE students is larger than 3.6 at the 0.10 significance level.
### Step 1: Formulate the Hypotheses
We are testing if the mean GPA is larger than 3.6. Therefore, our hypotheses are:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 3.6\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu > 3.6\)[/tex]
This is a right-tailed test because we are checking if the mean GPA is significantly greater than 3.6.
### Step 2: Identify the Test Statistic
Given:
- Sample size ([tex]\(n\)[/tex]) = 30
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 3.64
- Population mean ([tex]\(\mu\)[/tex]) = 3.6
- Sample standard deviation ([tex]\(s\)[/tex]) = 0.08
- Significance level ([tex]\(\alpha\)[/tex]) = 0.10
First, calculate the standard error of the mean (SEM):
[tex]\[
\text{Standard Error} = \frac{s}{\sqrt{n}} = \frac{0.08}{\sqrt{30}}
\][/tex]
Next, calculate the z-score (test statistic):
[tex]\[
z = \frac{\bar{x} - \mu}{\text{Standard Error}} = \frac{3.64 - 3.6}{\text{Standard Error}}
\][/tex]
The calculated test statistic is:
[tex]\[
z = 2.739
\][/tex]
### Step 3: Determine the Critical Value
For a right-tailed test at the 0.10 significance level ([tex]\(\alpha = 0.10\)[/tex]), the critical value can be found using the standard normal distribution (z-table). The critical z-value for [tex]\(\alpha = 0.10\)[/tex] is approximately 1.2816.
### Step 4: Make the Decision
Compare the test statistic to the critical value:
- If [tex]\(z > 1.2816\)[/tex], we reject the null hypothesis.
- If [tex]\(z \leq 1.2816\)[/tex], we fail to reject the null hypothesis.
Given [tex]\(z = 2.739\)[/tex]:
[tex]\[
2.739 > 1.2816
\][/tex]
### Conclusion:
Based on our calculated test statistic of 2.739, which is greater than the critical value of 1.2816, we reject the null hypothesis.
Therefore, we have sufficient evidence at the 0.10 significance level to support the claim that the mean GPA of ASBE students is larger than 3.6.
