Question

the mean age of graduate students at a university is at most 31 years with a standard deviation of two years. a random sample of 15 graduate students is taken. the sample mean is 32 years and the sample standard deviation is three years. are the data significant at the 1% level? the p - value is 0.0264. state the null and alternative hypotheses and interpret the p - value. h₀: μ = 31 hₐ: μ > 31 the p - value is 0.0264, so the data are not significant at the 1% level. h₀: μ ≤ 31 hₐ: μ > 31 the p - value is 0.0264, so the data are significant at the 1% level. h₀: μ ≤ 31 hₐ: μ > 31 the p - value is 0.0264, so the data are not significant at the 1% level.

Explanation:

• Null and Alternative Hypotheses:
• The null hypothesis \(H_0\) represents the status - quo or the claim we assume to be true initially. Here, the claim is that the mean age of graduate students is at most 31 years, so \(H_0:\mu\leq31\).
• The alternative hypothesis \(H_a\) is what we are trying to find evidence for. Since we want to check if the mean age is greater than 31 years (as the sample mean of 32 is greater than 31), \(H_a:\mu > 31\).
• p - value Interpretation:
• The p - value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.
• A significance level \(\alpha = 0.01\) (1% level). If \(p - value<\alpha\), we reject the null hypothesis. If \(p - value\geq\alpha\), we fail to reject the null hypothesis.
• Given \(p - value=0.0264\) and \(\alpha = 0.01\), since \(0.0264>0.01\), the data are not significant at the 1% level.

Answer:

\(H_0:\mu\leq31\), \(H_a:\mu > 31\). The data are not significant at the 1% level.