The average household size in a certain region several years...

The average household size in a certain region several years ago was 3.14 persons. A sociologist wishes to test, at the $5 \%$ level of significance, whether it is different now. Perform the test using the information collected by the sociologist: in a random sample of 75 households, the average size was 2.98 persons, with sample standard deviation 0.82 person.

Key Concepts

t-Distribution

The t-distribution is a probability distribution used in hypothesis testing when the sample size is relatively small or the population standard deviation is unknown. It accounts for additional uncertainty in the estimate of the standard deviation. When using sample data to estimate the population mean, the t-distribution is particularly useful in calculating the test statistic and determining the corresponding p-value.

Test Statistic

The test statistic is a standardized value calculated from sample data that is used to decide whether to reject the null hypothesis. In tests for population means, it measures how many standard errors the sample mean is away from the hypothesized population mean. The test statistic is then compared to a critical value from an appropriate probability distribution to assess the evidence against the null hypothesis.

p-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed in the sample data, assuming that the null hypothesis is true. It serves as a measure of the evidence against the null hypothesis. A small p-value indicates that the observed data are unlikely under the null hypothesis, leading researchers to consider rejecting the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement of no change or no effect, and it serves as the default assumption for the test. The alternative hypothesis (H?) posits that there is a significant difference or effect relative to the null hypothesis. In hypothesis testing, the goal is to determine whether the sample data provide sufficient evidence to reject the null hypothesis in favor of the alternative.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. It involves setting up a null hypothesis, which represents a baseline claim to be tested, and an alternative hypothesis, which is a statement that there is an effect or difference present. The process evaluates whether the data collected are consistent with the null hypothesis or if there is enough evidence to support the alternative hypothesis.

Significance Level

The significance level, often denoted by alpha (?), is the probability threshold set by the researcher for rejecting the null hypothesis. It represents the maximum acceptable risk of committing a Type I error, which is rejecting the null hypothesis when it is actually true. A commonly used significance level is 5%, which implies a 5% risk of incorrectly rejecting the null hypothesis.

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