Question

Suppose we have conducted a \(t\) test, with \(\alpha=0.05\), and the \(P\) -value is \(0.03 .\) For each of the following statements, say whether the statement is true or false and explain why. (a) We reject \(H_{0}\) with \(\alpha=0.05\). (b) We have significant evidence for \(H_{A}\) with \(\alpha=0.05\). (c) We would reject \(H_{0}\) if \(\alpha\) were \(0.10 .\) (d) We do not have significant evidence for \(H_{A}\) with \(\alpha=0.10\). (e) If \(H_{0}\) is true, the probability of getting a test statistic at least as extreme as the value of the \(t_{s}\) that was actually obtained is \(3 \%\). (f) There is a \(3 \%\) probability that \(H_{0}\) is true.

Short answer

True for statements (a), (b), (c), and (e), and false for statements (d) and (f).

Step-by-step solution

Understanding the concept of P-value and significance level

The P-value is the probability of obtaining test results at least as extreme as the observed results, under the assumption that the null hypothesis (H_{0}) is true. The significance level (alpha) is a threshold chosen by the researcher that determines when the null hypothesis should be rejected. If the P-value is less than alpha, we reject the null hypothesis.

Statement (a) evaluation

Since the P-value of 0.03 is less than the significance level of 0.05 (P < alpha), we reject the null hypothesis (H_{0}). Thus, the statement is true.

Statement (b) evaluation

Rejecting the null hypothesis (H_{0}) implies there is significant evidence in favor of the alternative hypothesis (H_{A}). Therefore, the statement is true, given that the P-value of 0.03 is less than alpha = 0.05.

Statement (c) evaluation

If we increase the significance level to alpha = 0.10, the P-value of 0.03 is still less than the new threshold. We would still reject the null hypothesis, making this statement true as well.

Statement (d) evaluation

Since we would reject the null hypothesis (H_{0}) for alpha = 0.10, as concluded in statement (c), this means we do have significant evidence for the alternative hypothesis (H_{A}). Therefore, the statement is false.

Statement (e) evaluation

This statement correctly describes the P-value. The P-value (0.03 or 3%) is indeed the probability of obtaining a test statistic as extreme as the observed, if the null hypothesis is true. Therefore, this statement is true.

Statement (f) evaluation

This statement is a common misconception about P-values. The P-value does not represent the probability that the null hypothesis is true. Instead, it measures the probability of observing the test statistic given the null hypothesis is true. Therefore, the statement is false.

Key concepts

Statistical Significance

Statistical significance plays a crucial role in hypothesis testing, indicating whether an observed effect is likely due to chance or not. It's a determination made by comparing the P-value to the significance level (alpha). When the P-value is lower than the preset alpha level, the results are said to be statistically significant.

This significance tells us that the findings are unlikely to have occurred if the null hypothesis were true. In our exercise's context, with a P-value of 0.03, which is less than alpha 0.05, the result is statistically significant, supporting the claim that the null hypothesis should be rejected in favor of the alternative hypothesis.

Null Hypothesis (H0)

The null hypothesis, denoted as H0, is a statement of 'no effect' or 'no difference.' It's the default assumption that there is no significant relationship or change due to the treatment or conditions tested in the study. In hypothesis testing, our goal is to determine whether the data provides enough evidence to reject H0 in favor of the alternative hypothesis, HA.

In the provided exercise, when we say that H0 is rejected, it means the evidence from our data suggests there is an effect or difference, and the observed results are statistically significant. The P-value assists in making this determination because it quantifies the probability of observing the results if H0 were true.

Alternative Hypothesis (HA)

The alternative hypothesis, represented as HA, is what a researcher seeks to support. It is a statement that proposes a statistical relationship, effect, or difference that contradicts the null hypothesis. When we reject H0 due to a low P-value, we lend support to HA, suggesting that the effect we are testing for is present.

In the context of the exercise, with a P-value of 0.03, we accept HA at the 0.05 alpha level, meaning that there is significant evidence of an effect or difference. However, it's important to note that while rejecting H0 supports HA, it does not 'prove' HA to be true; it simply implies the data are more consistent with HA than with H0.

Significance Level (alpha)

The significance level, denoted as alpha, is a critical threshold in hypothesis testing. It is a value set by the researcher before conducting the test, which determines the cutoff for rejecting the null hypothesis. Common alpha levels include 0.05, 0.01, and 0.10, representing a 5%, 1%, and 10% risk of rejecting a true null hypothesis, respectively.

In our exercise, an alpha of 0.05 means we have a 5% willing risk of a 'false positive' conclusion. However, if we adjust the alpha to 0.10, our willingness to accept error increases, making it easier to reject H0, as evidenced in statements (c) and (d). The chosen level of alpha influences the conservativeness of the test; lower alphas reduce the risk of falsely rejecting H0 but also make it harder to detect real differences.