The chamber of commerce of a Florida Gulf Coast community...

The chamber of commerce of a Florida Gulf Coast community advertises that area residenthal property is available at a mean cost of $$\$ 125,000$$ or less per lot. Suppose a sample of 32 properties provided a sample mean of $$\$ 130,000$$ per lot and a sample standard deviation of $$\$ 12,500$$. Use a .05 level of significance to test the validity of the advertising claim.

Key Concepts

One-Tailed Test

A one-tailed test is used when the alternative hypothesis predicts a direction of deviation from the null hypothesis, meaning that the rejection region is entirely on one end of the distribution. This approach is appropriate when the research interest is solely in determining whether a parameter is either greater than or less than a specific value, rather than simply different.

Degrees of Freedom

Degrees of freedom are a parameter that reflects the number of independent values in a statistical calculation that are free to vary, given certain constraints. For a t-test involving a sample mean, the degrees of freedom are usually equal to the sample size minus one. The degrees of freedom affect the shape of the t-distribution used to find critical values and p-values.

Test Statistic

The test statistic is a calculated value that summarizes the data in relation to the null hypothesis. In a t-test, it is computed by taking the difference between the observed sample mean and the hypothesized population mean, and then dividing this difference by the standard error of the mean. This statistic helps in determining how extreme the sample data are under the assumption that the null hypothesis is true.

Level of Significance

The level of significance, denoted by alpha (?), is a threshold that defines the probability of committing a Type I error—rejecting the null hypothesis when it is actually true. It is used to determine the critical value(s) from the relevant statistical distribution, which in turn are used to decide whether the observed test statistic falls into the rejection region.

t-Test

A t-test is a form of hypothesis testing that is used when the population standard deviation is unknown and the sample size is relatively small. It relies on the t-distribution to evaluate the test statistic, which involves the difference between the sample mean and the hypothesized population mean, scaled by the sample's standard error.

Null and Alternative Hypotheses

The null hypothesis (H0) typically states that there is no effect or that a parameter equals a specific value, while the alternative hypothesis (Ha) represents the claim that the parameter differs in a specific direction. This setup is crucial for determining the outcome of a test, as the decision to reject or not reject the null hypothesis depends on the evidence provided by the data.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating two mutually exclusive statements: the null hypothesis, which represents the default or accepted claim, and the alternative hypothesis, which represents the claim being tested. Decisions are based on whether the provided sample data offer enough evidence to reject the null hypothesis in favor of the alternative.

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