Final answer:
It is true that as population variance increases, the possibility of rejecting the null hypothesis also increases, because increased variance often results in a larger standard error, wider confidence intervals, and potentially a lower p-value leading to rejection of the null.
Step-by-step explanation:
The statement is true: as population variance increases, the possibility of rejecting the null hypothesis increases. When conducting a hypothesis test, an increased variance in the population implies that there is more spread in the data. This can lead to a larger standard error of the mean, assuming the sample size remains constant. However, this subsequently causes the confidence interval to widen, which may make it more likely to capture the true population mean if the null hypothesis is false. This is consistent with the central limit theorem, which states that as the sample size increases, the sampling distribution becomes more normal. When you compute a test statistic, a greater variance can inflate the test statistic value, which can decrease the p-value. If the p-value is lower than the predetermined significance level (α), you would reject the null hypothesis.
Making the decision on whether to reject or not is based on the comparison of α and the p-value. If α is greater than the p-value (α > p-value), then you reject the null hypothesis. This would be the case if you find that the variance is sufficiently large to suggest a statistically significant difference from what would be expected under the null hypothesis. For instance, reject the null hypothesis that the percentage of adults who have jobs is at least 88 percent if that percentage is observed to be lower with statistical significance.