Question

Match the four \(\mathrm{p}\) -values with the appropriate conclusion: (a) The evidence against the null hypothesis is significant, but only at the \(10 \%\) level. (b) The evidence against the null and in favor of the alternative is very strong. (c) There is not enough evidence to reject the null hypothesis, even at the \(10 \%\) level. (d) The result is significant at a \(5 \%\) level but not at a \(1 \%\) level. I. 0.0875 II. 0.5457 III. 0.0217 IV. \(\quad 0.00003\)

Short answer

The appropriate conclusions matched with the given p-values are: (a) matches with II. 0.0875, (b) matches with IV. 0.00003, (c) matches with II. 0.5457 and (d) matches with III. 0.0217.

Step-by-step solution

Match to Statement (a)

In statement (a), the evidence is significant at the 10% level. This implies a p-value less than 0.10 but greater than 0.05 (so it's not significant at the 5% level). From the options given, the corresponding p-value is 0.0875. Hence, II. 0.0875 matches with (a).

Match to Statement (b)

In statement (b), the evidence against the null hypothesis is very strong. Such strong evidence will be associated with a very low p-value. The smallest p-value offered here is IV. 0.00003. Hence, IV. 0.00003 matches with (b).

Match to Statement (c)

In statement (c), there is not enough evidence to reject the null hypothesis even at the 10% level, meaning the p-value here should be greater than 0.10. The only option available that fits this criterion is II. 0.5457. Hence, II. 0.5457 matches with (c).

Match to Statement (d)

In statement (d), a significant result at a 5% level but not the 1% level implies that the p-value is less than 0.05 and greater than 0.01. Therefore, the only remaining p-value, III. 0.0217, matches with (d).

Key concepts

Null Hypothesis Significance Testing

Understanding null hypothesis significance testing is crucial for interpreting p-values. The null hypothesis, often represented by the symbol H0, is a general statement or default position that there is no relationship between two measured phenomena. In the context of statistical testing, when we perform an experiment or a study, we measure this through a p-value, which tells us the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is correct.

When performing the test, if the p-value is low, it suggests that the observed data are unlikely if the null hypothesis were true, and thus, provides evidence against the null hypothesis. For example, in a clinical trial testing a new drug, the null hypothesis might be that the new drug has no effect on patients, compared to a placebo. If the results show a very low p-value, it would suggest that the effect observed (or even a more extreme one) is improbable due to chance alone, hinting that the drug may indeed have an effect.

In practice, we use a threshold, or alpha value, to decide whether to reject the null hypothesis. If the p-value is less than the chosen threshold, the null hypothesis is rejected in favor of an alternative hypothesis (H1 or Ha), which is the opposite of the null hypothesis. The example with the p-value of 0.0875 in the exercise above falls into a situation where the evidence against the null hypothesis is significant, but only at a less stringent level of 10%.

Statistical Evidence

Statistical evidence refers to the strength of the data in supporting or refuting a hypothesis. In hypothesis testing, the p-value quantifies this evidence. A low p-value indicates strong evidence against the null hypothesis, while a high p-value suggests weak evidence.

It's important to note that while the p-value can inform us about the strength of the evidence, it does not measure the probability that the hypothesis is true or false. For example, a p-value of 0.5457, as seen with statement (c) in the exercise, indicates that there is a 54.57% chance of observing the test results, or something more extreme, if the null hypothesis is true. This high p-value indicates there is not enough evidence to reject the null hypothesis.

Understanding statistical evidence is essential in research as it helps prevent incorrect conclusions. Strong evidence against the null hypothesis, like a p-value of 0.00003, leads to greater confidence in the validity of the research findings, potentially influencing further scientific inquiry or real-world decisions.

Hypothesis Testing Thresholds

Hypothesis testing thresholds are predetermined levels of significance used to decide whether to reject the null hypothesis. Commonly used thresholds or alpha levels are 0.05 (5%), 0.01 (1%), and 0.10 (10%). These values denote the probability of incorrectly rejecting a true null hypothesis, also known as the Type I error rate.

Choosing an appropriate threshold is a critical part of the experimental design and depends on the field of study, the standard of evidence required, and the potential consequences of making a Type I error. For example, higher stakes decisions often require a lower threshold (such as 1%) to minimize the chance of error. Conversely, in exploratory research, a higher threshold (such as 10%) might be acceptable.

The exercise provided illustrates different outcomes based on these thresholds. In statement (d), a result was significant at the 5% threshold (p-value less than 0.05) but not at the stricter 1% threshold (p-value still greater than 0.01). This differentiation emphasizes that researchers must consider the proper threshold for their specific scientific questions and the implications of their hypothesis tests.