The mean age of De Anza College students in a previous term...

The mean age of De Anza College students in a previous term was 26.6 years old. An instructor thinks the mean age for online students is older than $26.6 .$ She randomly surveys 56 online students and finds that the sample mean is 29.4 with a standard deviation of $2.1 .$ Conduct a hypothesis test.

Key Concepts

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to determine whether a statement about a population parameter is supported by sample data. It involves formulating two mutually exclusive hypotheses—the null hypothesis, which represents a baseline or status quo assumption, and the alternative hypothesis, which represents the claim or effect that is being tested. The process involves calculating a test statistic and comparing it with a critical value or using a p-value to decide whether to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H0) typically states that there is no effect or no difference, serving as a default or benchmark condition. The alternative hypothesis (Ha) reflects the researcher's actual claim — an effect or a difference exists. In hypothesis testing, the aim is to see if the sample data provides enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

Test Statistic

The test statistic is a standardized value calculated from sample data that measures the degree of divergence from what is expected under the null hypothesis. It incorporates the sample mean, sample size, and sample standard deviation (or population standard deviation, if known) to quantify the difference between the observed sample statistic and the hypothesized population parameter. This statistic is then compared to a critical value from a relevant probability distribution (such as the t-distribution or normal distribution) to facilitate decision-making.

Significance Level

The significance level, often represented by alpha (?), is a threshold set by the researcher that defines the probability of committing a Type I error—rejecting a true null hypothesis. Common choices for significance levels include 0.05, 0.01, and 0.10. It plays a crucial role in judging whether the observed test statistic is extreme enough to reject the null hypothesis.

P-value

The p-value is the probability of obtaining a test statistic that is as extreme as, or more extreme than, the observed value assuming that the null hypothesis is true. It quantifies the strength of evidence against the null hypothesis. A small p-value, typically less than the chosen significance level, indicates strong evidence against the null hypothesis and supports rejecting it in favor of the alternative hypothesis.

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