A random sample of 49 students indicates these students had a mean grade of 73 points on a final exam, with a standard deviation of 8 points. With a significance level of 5%, is there sufficient evidence to support the claim that the population mean is:
A. equal to 70 points?
B. greater than 75 points?
C. less than 80 points?
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To answer these questions, we need to perform a hypothesis test for each claim.
A. Equal to 70 points?
Null hypothesis (H0): μ = 70
Alternative hypothesis (H1): μ ≠ 70
The test statistic is calculated as follows:
Z = (X - μ) / (σ/√n) = (73 - 70) / (8/√49) = 2.625
The critical value for a two-tailed test at the 5% significance level is ±1.96. Since 2.625 > 1.96, we reject the null hypothesis. There is sufficient evidence to support the claim that the population mean is not equal to 70 points.
B. Greater than 75 points?
Null hypothesis (H0): μ = 75
Alternative hypothesis (H1): μ > 75
The test statistic is calculated as follows:
Z = (X - μ) / (σ/√n) = (73 - 75) / (8/√49) = -1.75
The critical value for a one-tailed test at the 5% significance level is -1.645. Since -1.75 < -1.645, we do not reject the null hypothesis. There is not sufficient evidence to support the claim that the population mean is greater than 75 points.
C. Less than 80 points?
Null hypothesis (H0): μ = 80
Alternative hypothesis (H1): μ < 80
The test statistic is calculated as follows:
Z = (X - μ) / (σ/√n) = (73 - 80) / (8/√49) = -6.125
The critical value for a one-tailed test at the 5% significance level is -1.645. Since -6.125 < -1.645, we reject the null hypothesis. There is sufficient evidence to support the claim that the population mean is less than 80 points.
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