Understanding P-values and Null Hypothesis Acceptance

In statistical hypothesis testing, the p-value is a crucial concept. It helps us decide whether to reject or fail to reject (accept) the null hypothesis based on the observed data.

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming that the null hypothesis ($H_0$) is true. A small p-value indicates strong evidence against the null hypothesis, while a large p-value indicates weak evidence against it.

The decision rule for hypothesis testing is typically based on comparing the p-value to a predetermined significance level, denoted by $\alpha$. Common values for $\alpha$ are 0.05 (5%), 0.01 (1%), or 0.10 (10%).

• If p-value $\leq \alpha$: We reject the null hypothesis ($H_0$). This suggests that the observed data is unlikely if the null hypothesis were true.
• If p-value > $\alpha$: We fail to reject (or accept) the null hypothesis ($H_0$). This suggests that the observed data is reasonably likely if the null hypothesis were true, and there is not enough evidence to conclude otherwise.

The question asks for p-values for which a null hypothesis is likely to be accepted. Based on the decision rule, this happens when the p-value is greater than the significance level ($\alpha$). Therefore, higher p-values are more likely to lead to the acceptance of the null hypothesis for standard significance levels.

Analyzing the Given P-values

Let's examine the given p-value options:

• A. 0.32 of 2%: This phrasing is unusual. If interpreted as a p-value of 0.32 and a significance level of 2% (0.02), then p-value (0.32) > $\alpha$ (0.02), leading to acceptance. If interpreted as 0.32 multiplied by 2%, the p-value is $0.32 \times 0.02 = 0.0064$. A p-value of 0.0064 is very low and would typically lead to rejection for standard $\alpha$ values like 0.05 or 0.01. Given the correct option, it's more likely the question intends to present a relatively high p-value here, potentially like 0.32.
• B. 32%: This is a p-value of 0.32. Comparing this to common significance levels:
  • If $\alpha = 0.05$, p-value (0.32) > $\alpha$ (0.05). $H_0$ is accepted.
  • If $\alpha = 0.01$, p-value (0.32) > $\alpha$ (0.01). $H_0$ is accepted.
  A p-value of 0.32 is significantly larger than typical $\alpha$ levels, making the acceptance of the null hypothesis very likely.
• C. 2%: This is a p-value of 0.02. Comparing this to common significance levels:
  • If $\alpha = 0.05$, p-value (0.02) $\leq \alpha$ (0.05). $H_0$ is rejected.
  • If $\alpha = 0.01$, p-value (0.02) > $\alpha$ (0.01). $H_0$ is accepted.
  A p-value of 0.02 is borderline. It would lead to rejection at the 5% significance level but acceptance at the 1% significance level. While acceptance is possible depending on $\alpha$, it's not as consistently "likely" as higher p-values across standard $\alpha$ levels.
• D. 0.42: This is a p-value of 0.42. Comparing this to common significance levels:
  • If $\alpha = 0.05$, p-value (0.42) > $\alpha$ (0.05). $H_0$ is accepted.
  • If $\alpha = 0.01$, p-value (0.42) > $\alpha$ (0.01). $H_0$ is accepted.
  A p-value of 0.42 is very high, making the acceptance of the null hypothesis very likely for any typical significance level.

Conclusion

Based on the analysis, p-values of 32% (0.32) and 0.42 are clearly much larger than common significance levels (like 0.05 or 0.01), leading to a high likelihood of accepting the null hypothesis. The p-value of 2% (0.02) is borderline and depends on the chosen $\alpha$. The interpretation of "0.32 of 2%" is ambiguous, but a p-value of 0.32 itself would also lead to likely acceptance.

Considering the options provided and the general principle that higher p-values mean weaker evidence against the null hypothesis and thus more likely acceptance, options B (32%) and D (0.42) represent p-values that are consistently high enough to lead to the acceptance of the null hypothesis across typical significance levels.

Therefore, the p-values for which a null hypothesis is likely to be accepted are B (32%) and D (0.42).

Revision Table: Key Hypothesis Testing Concepts

Additional Information: Hypothesis Testing Errors

When making a decision in hypothesis testing, there are two types of errors we might make:

• Type I Error: Rejecting the null hypothesis when it is actually true. The probability of making a Type I error is equal to the significance level $\alpha$. Reducing $\alpha$ (e.g., from 0.05 to 0.01) decreases the chance of a Type I error, but increases the chance of a Type II error.
• Type II Error: Failing to reject the null hypothesis when it is actually false. The probability of making a Type II error is denoted by $\beta$. The power of a test (1 - $\beta$) is the probability of correctly rejecting a false null hypothesis.

The choice of significance level $\alpha$ reflects the balance between these two types of errors. A lower $\alpha$ makes it harder to reject the null hypothesis, reducing Type I errors but increasing Type II errors. Conversely, a higher $\alpha$ makes it easier to reject the null hypothesis, increasing Type I errors but reducing Type II errors.