91Ó°ÊÓ

In hypothesis testing, the choice of distribution hinges on the sample size and the information available about the population. The z-distribution is a kind of normal distribution that plays an important role when testing hypotheses with large samples. It is particularly useful when the sample size is greater than 30, or when the population standard deviation is known. However, in many practical scenarios, we approximate the population distribution using the sample data.
A z-distribution is defined by:

• a mean of zero
• a standard deviation of one

This standard normal distribution helps standardize your data so you can better understand where your sample mean lies relative to the population mean. In the example problem, even though the population standard deviation is not given, a large sample size allows us to approximate the distribution using a z-distribution. This is why we calculate a z-statistic to determine if the sample mean of $16,377 is significantly different from the null hypothesis mean of $15,706.

The p-value is a fundamental concept in statistical hypothesis testing. It represents the probability of obtaining a test statistic as extreme as the observed one, assuming that the null hypothesis is true. It essentially helps us determine the evidence against the null hypothesis.
Calculating the p-value involves comparing the observed data with what would be expected under the null hypothesis. If the p-value is very low, it suggests that such extreme test results would be uncommon, leading us to question the null hypothesis. For this hypothesis test, we calculate a p-value from the z-statistic determined earlier.
In the context of our original exercise, a low p-value (less than the significance level of 1%) indicates that the observed sample mean is significantly different from the hypothesized population mean, thus providing evidence to reject the null hypothesis.

When deciding whether to reject a null hypothesis, we use the critical value approach as one of the tools. The critical value is a point in the distribution of the test statistic that is compared to the calculated statistic. Depending on the significance level and the direction of the test (one-tailed or two-tailed), this value helps define the threshold for decision-making.
To find the critical value for a specific significance level, you often use a z-table. In our exercise, with a 1% level of significance for a one-tailed test, the critical value separates the region where we reject the null hypothesis from where we do not. If the test statistic exceeds this critical value, it falls into the rejection region, indicating that the null hypothesis is unlikely to be true. In our problem, since we’re looking at a right-tailed test, if our calculated z-value is greater than the critical value, we reject the null hypothesis.

The null hypothesis is a statement used in hypothesis testing that asserts no effect or no difference exists. It's what you seek to test against with your data. In the context of our problem, the null hypothesis posits that the mean credit card debt of American households has not increased since 2015 and remains at $15,706.
Symbolically, the null hypothesis can be represented as:

• \( H_0: \mu = 15,706 \)

By setting up the null hypothesis, we frame the statistical problem and prepare to use our data to test this claim. A critical part of this process is determining the evidence we would need to reject it. The hypothesis testing process involves looking at how likely it is, under this hypothesis, for the observed sample statistics to occur. If it seems improbable, we may reject the null hypothesis and consider an alternative hypothesis, suggesting the mean debt has indeed increased.