Confidence Interval and Null Hypothesis

C

The confidence interval specifies the range over which the true value of the estimated parameter can lie without rejecting the null hypothesis at the given level of significance.

The correct answer is C. should be rejected at the corresponding significance level.

Let's break down the explanation:

In hypothesis testing, we start with a null hypothesis (H0) that represents the status quo or a claim that we want to test. The null hypothesis typically assumes that there is no significant difference or relationship between variables or that a parameter takes a specific value.

To test the null hypothesis, we collect sample data and calculate a test statistic, which measures the discrepancy between the observed data and what we would expect if the null hypothesis were true. We then compare this test statistic to a critical value or calculate a p-value.

A confidence interval is a range of values constructed from the sample data that is likely to contain the true value of the parameter we are interested in. It provides an estimate of the parameter's range of plausible values.

Now, if the hypothesized value of the parameter under the null hypothesis lies outside the confidence interval, it means that the hypothesized value is not within the plausible range of values based on the sample data. This suggests that the hypothesized value is unlikely to be true, and there is evidence against the null hypothesis.

In hypothesis testing, we have a predetermined significance level (often denoted as α) that represents the probability of rejecting the null hypothesis when it is actually true. If the hypothesized value lies outside the confidence interval, it indicates that the observed data are unlikely to occur under the null hypothesis, leading us to reject the null hypothesis. Therefore, the correct answer is C. should be rejected at the corresponding significance level.

Option A (none of these answers) is incorrect because there is a clear course of action when the hypothesized value lies outside the confidence interval.

Option B (cannot be rejected at the corresponding significance level) is incorrect because the evidence provided by the confidence interval suggests that the null hypothesis should be rejected.

Option D (is ill-specified) is incorrect because the null hypothesis being outside the confidence interval does not necessarily imply that it is ill-specified. It simply suggests that the hypothesized value is unlikely based on the sample data.