The mean age of graduate students at a University is at most...

The mean age of graduate students at a University is at most 31 y ears 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.

Key Concepts

Significance Level

The significance level, often denoted by alpha, is the threshold used to determine whether a p-value is sufficiently low to reject the null hypothesis. Commonly set at levels such as 1% or 5%, it represents the maximum probability of committing a Type I error, which is rejecting a true null hypothesis.

Interpretation of the P-value

Interpreting the p-value involves comparing it to the predetermined significance level. A small p-value (less than the significance level) indicates that the observed data are unlikely under the null hypothesis, providing evidence to reject it. Conversely, a p-value higher than the significance level suggests that the observed data do not provide sufficient evidence to reject the null hypothesis.

P-value

The p-value measures the probability of obtaining a test statistic as extreme as the one observed, or more extreme, under the assumption that the null hypothesis is true. It quantifies how consistent the data are with the null hypothesis, with lower values indicating stronger evidence against it.

Null Hypothesis

The null hypothesis is a statement asserting that there is no significant difference or effect, or that any observed difference is due to random chance. In the context of testing a mean, it often claims that the population mean is equal to a specified value or within an expected range.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about the properties of a population based on sample data. It involves establishing a null hypothesis, which represents a default position or a statement of no effect, and an alternative hypothesis, which represents a new claim or effect. Through this process, we assess whether the observed data provide sufficient evidence to reject the null hypothesis in favor of the alternative.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis. It indicates that the population parameter (such as the mean) is not equal to the specified value in the null hypothesis, implying there is an effect or difference that is statistically significant.

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