According to the Census Bureau, 3.13 people reside in the...

According to the Census Bureau, 3.13 people reside in the typical American household. A sample of 25 households in Arizona retirement communities showed the mean number of residents per household was 2.86 residents. The standard deviation of this sample was 1.20 residents. At the .05 significance level, is it reasonable to conclude the mean number of residents in the retirement community household is less than 3.13 persons?

Key Concepts

One-Sample t-Test

The one-sample t-test is used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean when the population standard deviation is unknown. Since the sample size is small (n = 25) and the standard deviation is estimated from the sample, the t-test is appropriate for this analysis.

p-Value

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the value observed, assuming that the null hypothesis is true. It quantifies the evidence against the null hypothesis. If the p-value is less than the significance level (0.05 in this case), the null hypothesis is rejected, suggesting that the sample provides sufficient evidence in support of the alternative hypothesis.

Degrees of Freedom

Degrees of freedom are related to the number of independent observations used to estimate a parameter, and for a one-sample t-test, it is equal to the sample size minus one. In this case, with 25 households, the degrees of freedom would be 24. This value is used to determine the critical value or p-value from the t-distribution.

Test Statistic

The test statistic for a one-sample t-test is calculated by taking the difference between the sample mean and the hypothesized population mean, and then dividing this difference by the standard error of the sample mean. The resulting t-value measures how many standard errors the sample mean is away from the hypothesized mean and is used to determine the significance of the results.

Significance Level

The significance level, denoted by alpha, is the threshold probability for rejecting the null hypothesis. It represents the maximum risk of committing a Type I error (incorrectly rejecting a true null hypothesis) that the researcher is willing to accept. A significance level of 0.05 implies that there is a 5% chance of rejecting the null hypothesis when it is, in fact, true.

Null and Alternative Hypotheses

The null hypothesis (H0) is a statement asserting that there is no difference or no effect, while the alternative hypothesis (H1) indicates the presence of an effect or a difference. In this context, the null hypothesis would state that the mean number of residents per household in the retirement community is 3.13, and the alternative hypothesis would assert that it is less than 3.13.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population parameter based on sample data. It involves formulating two competing hypotheses – a null hypothesis that represents the status quo or no effect, and an alternative hypothesis that represents the research claim – and then using sample evidence to assess whether there is enough evidence to reject the null hypothesis in favor of the alternative.

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