Question

A confidence interval for a sample is given, followed by several hypotheses to test using that sample. In each case, use the confidence interval to give a conclusion of the test (if possible) and also state the significance level you are using. A \(90 \%\) confidence interval for \(p_{1}-p_{2}: 0.07\) to 0.18 (a) \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1} \neq p_{2}\) (b) \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}>p_{2}\) (c) \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}

Short answer

Based on the given 90% confidence interval, we reject the hypotheses \(H_{0}: p_{1}=p_{2}\) for alternatives \(H_{a}: p_{1} \neq p_{2}\) and \(H_{a}: p_{1}>p_{2}\), but fail to reject it for \(H_{a}: p_{1}<p_{2}\). The significance level used is 10%.

Step-by-step solution

Analysis of the 90% Confidence Interval

First of all, analyze the 90% confidence interval of \(p_{1}-p_{2}\) which ranges from 0.07 to 0.18. A \(90 \%\) confidence interval means that we can be \(90 \%\) confident that the true difference between \(p_{1}\) and \(p_{2}\) lies within this range.

Testing Hypothesis (a)

Test the hypothesis \(H_{0}: p_{1}=p_{2}\) against \(H_{a}: p_{1} \neq p_{2}\). If \(p_{1}=p_{2}\), then \(p_{1}-p_{2}=0\). However, 0 is not within the provided confidence interval of 0.07 to 0.18. Therefore, we reject \(H_{0}\) at the \(10 \%\) significance level.

Testing Hypothesis (b)

Now test the hypothesis \(H_{0}: p_{1}=p_{2}\) against \(H_{a}: p_{1}>p_{2}\). If \(p_{1}=p_{2}\), then \(p_{1}-p_{2}=0\). The entire confidence interval is above 0, so we reject \(H_{0}\) at the \(10 \%\) significance level.

Testing Hypothesis (c)

Lastly, test the hypothesis \(H_{0}: p_{1}=p_{2}\) against \(H_{a}: p_{1}<p_{2}\). If \(p_{1}<p_{2}\), then \(p_{1}-p_{2}<0\). The entire confidence interval is above 0, so we fail to reject \(H_{0}\) at the \(10 \%\) significance level.

Key concepts

Confidence Intervals

Confidence intervals are a range of values, derived from sample data, that are believed to contain the true parameter of the population with a certain probability. In hypothesis testing, they serve as an alternative way to test the hypothesis, especially when you do not have access to the original dataset or enough information to perform a standard test.

In the given exercise, a 90% confidence interval for the difference between two proportions, labeled as \( p_{1} \) and \( p_{2} \), was provided. The interval from 0.07 to 0.18 implies that we are 90% confident that the true difference between \( p_{1} \) and \( p_{2} \) lies within this range. By interpreting this confidence interval, one can draw conclusions about the plausibility of the null hypothesis (in this case, that \( p_{1} = p_{2} \) or \( p_{1}-p_{2} = 0 \) ). If the interval does not contain the value under the null hypothesis (0 in this scenario), we have evidence to reject the null hypothesis. Since the interval does not include 0, we can conclude that there is a statistically significant difference between \( p_{1} \) and \( p_{2} \).

Significance Level

The significance level, often denoted by \( \alpha \), is a threshold used to decide whether a statistical hypothesis should be rejected. It is the probability of making a Type I error, which occurs when the null hypothesis is true, but is incorrectly rejected.

In hypothesis testing, if the p-value is less than \( \alpha \), the null hypothesis is rejected. A common significance level used is 0.05, but in the provided exercise, a 90% confidence interval corresponds to a significance level of \( \alpha = 0.10 \). This means there is a 10% risk of wrongly rejecting the null hypothesis \( H_{0}: p_{1} = p_{2} \). When deciding whether to reject \( H_{0} \) based on the confidence interval, you're essentially testing it at the 10% significance level.

Difference Between Two Proportions

When dealing with two separate groups, researchers often want to compare the proportions between these groups, for example, the proportion of success in one group versus another. The difference between two proportions is a measure of how much the proportions in two groups differ from one another.

In this context, the hypothesis tests are structured to examine whether the proportions from two different populations, \( p_{1} \) and \( p_{2} \), are the same or not (null hypothesis) or whether one is larger or smaller than the other (alternative hypothesis). The exercise provided uses a 90% confidence interval to test three different hypotheses related to the difference between two proportions. Each hypothesis (a, b, and c) examines a different relationship (equal, greater than, or less than) between \( p_{1} \) and \( p_{2} \) and draws conclusions based on whether 0 is included in the confidence interval.