Question

Test the claim that the mean GPA of night students is significantly different than 3.1 at the 0.01 significance level.

The null and alternative hypothesis would be:

H0:μ=3.1H0:μ=3.1
H1:μ≠3.1H1:μ≠3.1

H0:μ=3.1H0:μ=3.1
H1:μ<3.1H1:μ<3.1

H0:p=0.775H0:p=0.775
H1:p>0.775H1:p>0.775

H0:μ=3.1H0:μ=3.1
H1:μ>3.1H1:μ>3.1

H0:p=0.775H0:p=0.775
H1:p<0.775H1:p<0.775

H0:p=0.775H0:p=0.775
H1:p≠0.775H1:p≠0.775

The test is:

two-tailed

left-tailed

right-tailed

Based on a sample of 65 people, the sample mean GPA was 3.14 with a standard deviation of 0.05

The test statistic is:  (to 2 decimals)

The positive critical value is:  (to 2 decimals)

Based on this we:

• Reject the null hypothesis
• Fail to reject the null hypothesis

Answer 1

claim: the mean GPA of night student is significantly different than 3.1 i.e μ ≠ 3.1

The following null and alternative hypotheses need to be tested:

Ho: μ = 3.1

Ha: μ ≠ 3.1 (claim)

This corresponds to a two-tailed test,

The provided sample mean is Xˉ=3.14 and the sample standard deviation is s=0.05, and the sample size is n=65.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ = 3.1

Ha: μ ≠ 3.1

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.01, and the critical value for a two-tailed test is

tc​=2.655.

The rejection region for this two-tailed test is R={t:∣t∣>2.655}

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣t∣=6.45>tc​=2.655, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0, and since p=0<0.01, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is different than 3.1, at the 0.01 significance level.

Confidence Interval

The 99% confidence interval is 3.124<μ<3.156.

Graphically

​​​​​Test statistic:

t=6.45

critical value:

Based on the information provided, the significance level is α=0.01, and the critical value for a two-tailed test is

tc​=2.655

Since it is observed that ∣t∣=6.45>tc​=2.655, it is then concluded that reject the null Hypothesis

Final conclusion

the mean GPA of night student is significantly different than 3.1

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