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26.06.2020 •
Business
A student believes that the average grade on the statistics final examination is 87. A sample of 36 final
examinations is taken. The average grade in the sample is 83.96. The population variance is 144.
a. State the null and alternative hypotheses.
b. Using a critical value, test the hypothesis at the 5% level of significance.
c. Using a p-value, test the hypothesis at the 5% level of significance.
d. Using a confidence interval, test the hypothesis at the 5% level of significance.
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Ответ:
(a) H₀: μ = 87 vs. Hₐ: μ ≠ 87.
(b) . The average grade on the statistics final examination is 87.
(c) The p-value = 0.1286 > α = 0.05. The average grade on the statistics final examination is 87.
(d) The 95% confidence interval for the average grade on the statistics final examination is (80.04, 87.88).
Explanation:
A statistical hypothesis test is to be performed to determine whether the average grade on the statistics final examination is 87.
(a)
The hypothesis can be defined as follows:
H₀: The average grade on the statistics final examination is 87, i.e. μ = 87.
Hₐ: The average grade on the statistics final examination is not 87, i.e. μ ≠ 87.
(b)
The information provided is:
As the population variance is provided, we will use a z-test for single mean.
Compute the test statistic value as follows:
The test statistic value is -1.52.
Decision rule:
Reject H₀ if:
The calculated value of .
The null hypothesis will not be rejected.
Conclusion:
The average grade on the statistics final examination is 87.
(c)
Decision Rule:
If the p-value of the test is less than the significance level then the null hypothesis will be rejected.
Compute the p-value for the two-tailed test as follows:
*Use a z-table for the probability.
The p-value of the test is 0.1286.
p-value = 0.1286 > α = 0.05
The null hypothesis will not be rejected.
Thus, it can be concluded that average grade on the statistics final examination is 87.
(d)
Compute the 95% confidence interval for the average grade on the statistics final examination as follows:
The 95% confidence interval for the average grade on the statistics final examination is (80.04, 87.88).
As the 95% confidence interval consists of the null value, i.e. 87, the null hypothesis will not be rejected.
Hence, concluding that the average grade on the statistics final examination is 87.
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