Question

Use the definition of the \(P\) -value to explain the following: a. Why \(H_{0}\) would be rejected if \(P\) -value \(=.0003\) b. Why \(H_{0}\) would not be rejected if \(P\) -value \(=.350\)

Short answer

The P-value quantifies the strength of evidence against the null hypothesis \(H_{0}\). If the P-value is below a predetermined significance level (often 0.05), then \(H_{0}\) is rejected. Thus, for a P-value of 0.0003 (< 0.05), \(H_{0}\) would be rejected. Conversely, for a P-value of 0.350 (> 0.05), there is not sufficient evidence to reject \(H_{0}\), so \(H_{0}\) is not rejected.

Step-by-step solution

Understanding the P-value

The P-value in a hypothesis testing scenario represents the probability of obtaining a test statistic result at least as extreme as the one that was actually observed, given that the null hypothesis is true. If this probability (the P-value) is smaller than a predetermined significance level (typically 0.05), then there is strong evidence against the null hypothesis, so it is rejected.

Application of P-value interpretation to part a

When the P-value is 0.0003, it means there's only a 0.03% chance of obtaining a test statistic as extreme as the observed one, assuming the null hypothesis is true. This is a very low probability, far below the typical 5% significance level. As such, the evidence against the null hypothesis is considered strong, and hence \(H_{0}\) would be rejected.

Application of P-value interpretation to part b

On the other hand, when the P-value is 0.350, it means there's a 35% chance of obtaining a test statistic as extreme as the observed one if the null hypothesis is true. This is a far higher probability, above the common 5% significance level. Therefore, the evidence against the null hypothesis is considered not strong enough to reject it, so \(H_{0}\) would not be rejected.

Key concepts

Null Hypothesis Rejection

The null hypothesis, often denoted by H_{0}, is a general statement or default position that there is no relationship between two measured phenomena or no association among groups. Rejecting the null hypothesis is a crucial step in hypothesis testing. It essentially means that you have sufficient evidence to conclude that the null hypothesis is not likely to be true.

In the context of the exercise, if the P-value is .0003, it indicates that the results observed in the study (or more extreme results) would occur only 0.03% of the time if the null hypothesis were true. Since this chance is extremely low, scientists have agreed on a threshold below which they conclude the null hypothesis is unlikely. This leads to its rejection, and it's typically done with a high level of confidence in the result.

Significance Level

The significance level, denoted by \(\beta\), is a threshold chosen by the researcher to decide whether to reject the null hypothesis. The most commonly used significance level is 0.05 or 5%. This standard has been established as a balance between being too lenient and too strict when it comes to identifying a statistically significant result.

The significance level is a critical value in statistical tests and represents the probability of rejecting the null hypothesis when it is actually true, an error known as a Type I error. When your P-value is lower than the significance level, it implies that the observed data are inconsistent with the assumption that the null hypothesis is true, favoring the alternative hypothesis.

Test Statistic

A test statistic is a value calculated from the sample data during a hypothesis test. Its purpose is to help determine the likelihood of the null hypothesis given the data. The test statistic can be a t-value, z-score, chi-squared statistic, or any number of other statistics, depending on the type of test being performed.

In practical terms, the test statistic compares your data to what is expected under the null hypothesis. The more the test statistic diverges from what we'd expect if \(H_{0}\) were true, the less likely the null hypothesis is accurate. The P-value directly relates to the test statistic, as it tells you the probability of seeing a value as extreme as or more extreme than the test statistic, assuming the null hypothesis is true.

Probability

In hypothesis testing, probability measures the likelihood of a certain event occurring. It ranges from 0 to 1, with lower values indicating a lower likelihood and higher values indicating a greater likelihood. When you calculate a P-value in the context of hypothesis testing, you're computing the probability of obtaining results at least as extreme as the observed results, under the assumption that the null hypothesis is correct.

The concept of probability is foundational when interpreting P-values. A P-value of .350 means there is a 35% chance of obtaining a test statistic as extreme as the observed one if the null hypothesis were true—a relatively high probability, which suggests that such an outcome isn't particularly unusual and, thus, doesn't provide strong evidence against the null hypothesis.