A recent article in The Wall Street Journal reported that...

A recent article in The Wall Street Journal reported that the home equity loan rate is now less than $4 \% .$ A sample of eight small banks in the Midwest revealed the following home equity loan rates (in percent): $\begin{array}{lllllll}3.6 & 4.1 & 5.3 & 3.6 & 4.9 & 4.6 & 5.0 & 4.4\end{array}$ At the .01 significance level, can we conclude that the home equity loan rate for small banks is less than $4 \%$ ? Estimate the $p$ -value.

A recent article in The Wall Street Journal reported that the home equity loan rate is now less than $4 \% .$ A sample of eight small banks in the Midwest revealed the following home equity loan rates (in percent):
$\begin{array}{lllllll}3.6 & 4.1 & 5.3 & 3.6 & 4.9 & 4.6 & 5.0 & 4.4\end{array}$
At the .01 significance level, can we conclude that the home equity loan rate for small banks is less than $4 \%$ ? Estimate the $p$ -value.

Key Concepts

Hypothesis Testing

Hypothesis testing is a method of making decisions using data from a sample. It involves stating a null hypothesis, which represents a default position or a baseline claim, and an alternative hypothesis, which is the claim being tested. The outcome of the test helps to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) is a statement that there is no effect or no difference, such as a population mean being equal to a specified value. The alternative hypothesis (H1) is the claim that researchers want to test, for example, asserting that the population mean is less than or greater than the specified value. Clearly defining these hypotheses is crucial for guiding the testing process.

One-sample t-test

The one-sample t-test is a statistical procedure used to determine whether the mean of a sample differs significantly from a known or hypothesized population mean. This test is especially useful when the sample size is small and the population standard deviation is unknown, as it accounts for the additional uncertainty by using the sample standard deviation in its calculations.

t-distribution

The t-distribution is a probability distribution that is used instead of the normal distribution when dealing with small sample sizes or when the population standard deviation is unknown. It has heavier tails than the normal distribution, which means it provides a more conservative estimate of extreme values. The shape of the t-distribution depends on the degrees of freedom associated with the sample.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation that are free to vary when estimating statistical parameters. In a one-sample t-test, the degrees of freedom are determined by the sample size minus one. This parameter influences the shape of the t-distribution and plays a role in determining the critical values for the test.

Significance Level

The significance level, often denoted as alpha (?), is the threshold used to decide whether a test result is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). Common significance levels are 0.05, 0.01, or 0.10, and in this context, a 0.01 level is used to make a more stringent decision.

P-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, given that the null hypothesis is true. It provides a measure of the strength of the evidence against the null hypothesis. A smaller p-value indicates stronger evidence to reject the null hypothesis, and comparing the p-value to the significance level helps decide whether to reject H0.

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