Question

Suppose we have conducted a \(t\) test, with \(\alpha=0.10\), and the \(P\) -value is \(0.07 .\) For each of the following statements, say whether the statement is true or false and explain why. (a) We reject \(H_{0}\) with \(\alpha=0.10\). (b) We have significant evidence for \(I I_{A}\) with \(\alpha=0.10\). (c) We would reject \(H_{0}\) if \(\alpha\) were \(0.05 .\) (d) We do not have significant evidence for \(H_{A}\) with \(\alpha=0.05\). (e) The probability that \(\bar{Y}_{1}\) is greater than \(\bar{Y}_{2}\) is \(0.07 .\)

Short answer

The answers are: (a) True, we reject \(H_0\) with \(\alpha=0.10\); (b) True, there is significant evidence for \(H_A\); (c) False, we would not reject \(H_0\) if \(\alpha\) were \(0.05\); (d) True, we do not have significant evidence for \(H_A\) with \(\alpha=0.05\); (e) False, the p-value does not indicate the probability of one mean being greater than the other.

Step-by-step solution

- Understanding Hypothesis Testing

In hypothesis testing, we compare the p-value to the significance level \(\alpha\) to decide whether to reject the null hypothesis \(H_0\). If the p-value is less than or equal to \(\alpha\), we reject \(H_0\); if it is greater, we fail to reject \(H_0\).

- Statement (a): Rejecting \(H_0\) with \(\alpha=0.10\)

Compare the p-value \(0.07\) with the significance level \(\alpha=0.10\). Since \(0.07 < 0.10\), we reject \(H_0\). Therefore, the statement is true.

- Statement (b): Significant Evidence for \(H_A\) with \(\alpha=0.10\)

Since we rejected \(H_0\) at \(\alpha=0.10\), we have significant evidence to support the alternative hypothesis \(H_A\). Therefore, the statement is true.

- Statement (c): Rejecting \(H_0\) with \(\alpha=0.05\)

Checking if \(H_0\) would be rejected with \(\alpha=0.05\), we find that \(0.07 > 0.05\). Hence, we would not reject \(H_0\) at this level of significance. The statement is false.

- Statement (d): Significant Evidence for \(H_A\) with \(\alpha=0.05\)

Given that the p-value is greater than \(\alpha=0.05\), we do not have significant evidence to support \(H_A\) at this level. The statement is true.

- Statement (e): Interpreting the P-value

The p-value of \(0.07\) represents the probability of observing a test statistic as extreme as the one observed under the assumption that the null hypothesis is true. It does not specify the probability that \(\bar{Y}_{1}\) is greater than \(\bar{Y}_{2}\). The statement is false.

Key concepts

P-value

The p-value is a crucial concept in hypothesis testing that tells us how likely it is to obtain a test statistic as extreme as the one observed, assuming that the null hypothesis is true. In simpler terms, it measures the strength of evidence against the null hypothesis.

For example, a p-value of 0.07, as in the exercise, suggests that there's a 7% chance of observing a test statistic at least as extreme as the one calculated if the null hypothesis is true. If this p-value is less than the predefined significance level, we say that the evidence is strong enough to reject the null hypothesis.

Interpreting the P-value Correctly

It's important to note that the p-value is not the probability that the null hypothesis is true or false, and it’s not the probability of making a mistake by rejecting the null hypothesis. Those common misconceptions lead to incorrect interpretations of statistical results.

Significance Level

The significance level, usually denoted by \( \alpha \), is a threshold chosen by the researcher to determine the cut-off point at which they will reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true, known as the Type I error rate.

Commonly used significance levels include 0.10, 0.05, and 0.01, marking the maximum acceptable probability of mistakenly rejecting the true null hypothesis. The smaller the value of \( \alpha \), the stronger the evidence must be to reject the null hypothesis.

For instance, the exercise mentions \( \alpha=0.10 \), meaning there's a 10% risk of a Type I error. If researchers are comfortable with this level of risk, they can reject the null hypothesis when the p-value is below 0.10.

Null Hypothesis

The null hypothesis (usually denoted \( H_0 \)) is a statement of no effect or no difference that serves as the starting point for statistical testing. It posits that any observed difference in data is due to chance or random variability rather than a true effect.

If testing the effectiveness of a new drug, for example, the null hypothesis might state that the drug has no effect on patients compared to a placebo.

Role in Hypothesis Testing

The role of the null hypothesis in testing is to create a benchmark for measuring how unusual the data are. Researchers try to disprove or reject the null hypothesis in favor of the alternative hypothesis by showing that the observed data would be very unlikely if the null hypothesis were true.

Alternative Hypothesis

The alternative hypothesis (denoted \( H_A \) or \( H_1 \) ) is a statement that the researcher seeks to support, suggesting that there is a statistically significant effect, or a difference, or a relationship. This hypothesis is considered only when the null hypothesis is rejected.

Returning to the drug example, the alternative hypothesis would state that the new drug does have a significant effect on patients compared to a placebo. It's what we suspect might be true and we aim to gather evidence via our statistical test to support this suspicion.

Formulating Hypotheses

In hypothesis testing, both the null and alternative hypotheses need to be clearly defined before any data collection or testing begins. This ensures that the test is objective and the conclusions are valid.

Statistical Significance

Statistical significance is the determination of whether the results obtained in a study are unlikely to have occurred purely by chance, given that the null hypothesis is true. This concept helps researchers assess the reliability of their results.

Results are considered statistically significant if the p-value is below the significance level. This means the data are sufficiently strong to reject the null hypothesis and to infer that the effect observed (or the difference) is not due to random variation.

Importance of Statistical Significance

Statistical significance does not imply that the results are practically significant or that they have a large effect size—it merely indicates that the findings are not consistent with the null hypothesis. In research, establishing statistical significance is often essential before making claims or drawing conclusions based on the data.