Question

What Gives a Small P-value? In each case below, two sets of data are given for a two-tail difference in means test. In each case, which version gives a smaller \(\mathrm{p}\) -value relative to the other? (a) Both options have the same standard deviations and same sample sizes but: Option 1 has: \(\quad \bar{x}_{1}=25 \quad \bar{x}_{2}=23\) $$ \text { Option } 2 \text { has: } \quad \bar{x}_{1}=25 \quad \bar{x}_{2}=11 $$ (b) Both options have the same means \(\left(\bar{x}_{1}=22,\right.\) \(\left.\bar{x}_{2}=17\right)\) and same sample sizes but: Option 1 has: \(\quad s_{1}=15 \quad s_{2}=14\) $$ \text { Option } 2 \text { has: } \quad s_{1}=3 \quad s_{2}=4 $$ (c) Both options have the same means \(\left(\bar{x}_{1}=22,\right.\) \(\left.\bar{x}_{2}=17\right)\) and same standard deviations but: Option 1 has: \(\quad n_{1}=800 \quad n_{2}=1000\) $$ \text { Option } 2 \text { has: } \quad n_{1}=25 \quad n_{2}=30 $$

Short answer

For each of the parts: (a) Option 2 will have a smaller p-value, (b) Option 2 will have a smaller p-value, and (c) Option 1 will have a smaller p-value.

Step-by-step solution

- Analyze the differences in means

In a two-tail difference of means test, a larger difference in sample means tend to produce smaller p-values. To see why, recall that a smaller p-value indicates stronger evidence against the null hypothesis, which in this case usually states that there's no difference between the means. So, a larger difference in means corresponds to a greater deviation from the null hypothesis, leading to smaller p-value. Therefore for part (a), Option 2 should produce a smaller p-value because the difference in means is larger than that of Option 1.

- Consider the effect of standard deviations

When the sample means and sizes are the same, a smaller standard deviation will result in a smaller p-value. This is because a smaller standard deviation indicates that the data points are closer to the mean, making the difference between the means appear more significant. That's why in part (b) Option 2 would have a smaller p-value. The standard deviations are smaller, so the difference in means will be more significant relative to the variability in the data.

- Evaluate the effect of sample sizes

Finally, larger sample sizes tend to produce smaller p-values. The reason is that having more data increases our ability to detect a difference where one exists. Hence, in part (c), Option 1 would have a smaller p-value because the sample sizes are much larger than those of Option 2.

Key concepts

Two-Tail Difference in Means Test

In statistical analysis, a two-tail difference in means test is a common procedure to determine if there is a significant difference between the means of two populations. This type of test is called 'two-tail' because it considers extreme values—and thus potential significant differences—in both tails of the distribution curves.

When conducting such a test, the p-value is a crucial statistic. It offers evidence against the null hypothesis, which typically posits no difference between the two means. A smaller p-value signifies more substantial evidence against the null hypothesis. A larger difference between the means usually yields a smaller p-value since it suggests that the likelihood of such an extreme observed result occurring by chance is low. This concept was exemplified in part (a) of the exercise where Option 2, with a larger difference in means, should produce a smaller p-value.

Standard Deviation

Understanding standard deviation is critical as it measures the amount of variability or dispersion within a set of data values. In other words, it indicates the extent to which the individual data points in a dataset deviate from the mean. Within the context of a two-tail difference in means test, a smaller standard deviation suggests that the data points are tightly grouped around the mean, entailing less variability.

Why does this matter for p-values? Lower variability strengthens the statistical evidence for a difference in means when the sample means are the same, as observed in part (b) of the problem. In that scenario, Option 2 has smaller standard deviations, so the differences between the means appear more remarkable relative to the consistency of the data, thereby resulting in a smaller p-value.

Sample Size

Sample size plays a fundamental role in the validity of statistical tests. A larger sample size allows for more precise estimation of the population parameters and enhances the robustness of the statistical test's results. When comparing two means, increasing the sample size can detect even small differences between the means, thereby lowering the p-value, indicating more substantial evidence against the null hypothesis of no difference.

As applied in part (c) of the exercise, large sample sizes as in Option 1 (with hundreds of samples) versus smaller ones in Option 2 (with around two dozen samples each) can have a significant effect on p-values. The larger sizes in Option 1 reduce the impact of random variability, making it easier to identify true differences, thereby producing a smaller p-value compared to Option 2's smaller sample sizes.