A recent survey by nerdwallet.com indicated Americans paid a...

A recent survey by nerdwallet.com indicated Americans paid a mean of $\$ 6,658$ interest on credit card debt in 2017 . A sample of 12 households with children revealed the following amounts. At the .05 significance level, is it reasonable to conclude that these households paid more interest? $\begin{array}{lllllllll}7,077 & 5,744 & 6,753 & 7,381 & 7,625 & 6,636 & 7,164 & 7,348 & 8,060 & 5,848 & 9,275 & 7,052\end{array}$

A recent survey by nerdwallet.com indicated Americans paid a mean of $\$ 6,658$ interest on credit card debt in 2017 . A sample of 12 households with children revealed the following amounts. At the .05 significance level, is it reasonable to conclude that these households paid more interest?
$\begin{array}{lllllllll}7,077 & 5,744 & 6,753 & 7,381 & 7,625 & 6,636 & 7,164 & 7,348 & 8,060 & 5,848 & 9,275 & 7,052\end{array}$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method that involves making an initial assumption (the null hypothesis) and then determining, based on sample data, whether this assumption can be rejected in favor of an alternative hypothesis. The process involves calculating a test statistic that is compared against a critical value or used to compute a p-value which helps in making an inference about the population.

One-sample t-test

The one-sample t-test is used when assessing whether the mean of a sample differs significantly from a known or hypothesized value, particularly when the population standard deviation is unknown and the sample size is small. It relies on the t-distribution to account for additional uncertainty from estimating the standard deviation from the sample data.

Null and Alternative Hypotheses

In the context of hypothesis testing, the null hypothesis represents a baseline claim (such as no difference or effect) whereas the alternative hypothesis represents a claim of interest (such as a difference or effect). Establishing these hypotheses is crucial as they guide the entire testing process, including determining the direction of the test (one-tailed or two-tailed).

Significance Level and Decision Rule

The significance level, often denoted by alpha (?), is the threshold used to determine when to reject the null hypothesis. It represents the maximum acceptable probability of making a Type I error (incorrectly rejecting a true null hypothesis). A decision rule is then applied whereby the computed p-value is compared to ?, or the test statistic is compared to a critical value, to decide whether to reject the null hypothesis.

p-value and Critical Values

The p-value is the probability of observing a test statistic as extreme as the one calculated from the sample data, assuming the null hypothesis is true. If this probability is less than or equal to the significance level, it provides sufficient evidence to reject the null hypothesis. Alternatively, critical values, derived from the t-distribution in a one-sample t-test, are used as cutoffs to determine the rejection region for the test statistic.

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