Question

In Exercises 6.156 to 6.161: (a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether there is a difference between two groups in the proportion who voted, if 45 out of a random sample of 70 in Group 1 voted and 56 out of a random sample of 100 in Group 2 voted.

Short answer

Based on the calculations, the test statistic will determine if there is a difference in the voting proportions of the two groups. Final verdict will depend upon comparison with critical value.

Step-by-step solution

Calculate the Proportions

Calculate the proportions of those who voted in each group. For Group 1, the proportion is \( \frac{45}{70} = 0.643 \). For Group 2, the proportion is \( \frac{56}{100} = 0.56 \).

Calculate the Pooled Proportion

The pooled proportion is calculated by dividing the total number of successes by the total sample size. The total number of successes is \( 45 + 56 = 101 \) and the total sample size is \( 70 + 100 = 170 \). So, the pooled proportion is \( \frac{101}{170} = 0.594 \).

Formulate the Hypotheses

The null hypothesis is that there is no difference between the proportions, which is written in mathematical terms as \( p1 = p2 \). The alternative hypothesis is that there is a difference in the proportions, which we write as \( p1 \neq p2 \).

Calculate the Test Statistic

Using the formula for the test statistic in a hypothesis test comparing two proportions, \( Z = \frac {(p1 - p2) - 0 }{ \sqrt {P(1 - P)(\frac{1}{n1} + \frac{1}{n2} )}} \) where \( p1 \) and \( p2 \) are the sample proportions, \( P \) is the pooled proportion, and \( n1 \) and \( n2 \) are the sample sizes. Plugging in the values gives \( \frac {(0.643 - 0.56) - 0 }{ \sqrt {0.594(1 - 0.594)(\frac{1}{70} + \frac{1}{100} )}} \). Calculate this to get the test statistic.

Analyze the Result

Compare the test statistic to a typically chosen significance level in a normal distribution, for example \( \alpha = 0.05 \). If the test statistic is greater than 1.96 or less than -1.96, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

Key concepts

Sample Proportions Calculation

Understanding the concept of sample proportions is vital in statistics, especially when comparing groups. The sample proportion is simply the fraction of successes (voted, in this case) in a sample. To calculate this, divide the number of individuals with the trait of interest by the total number in the sample. For instance, in Group 1, 45 out of 70 individuals voted, yielding a sample proportion of \( \frac{45}{70} = 0.643 \). Similarly, Group 2 had a sample proportion of \( \frac{56}{100} = 0.56 \).

Why are sample proportions important? They provide a snapshot of a specific characteristic within a sample, which can be used to infer about the larger population. Solid comprehension of sample proportions is the bedrock for further comparative analysis between groups.

Pooled Proportion Calculation

When comparing two proportions, statisticians often use the pooled proportion. This is a weighted average of the sample proportions that assumes the null hypothesis of no difference between the group proportions is true. To find it, you combine the successes of the groups and divide by the combined sample sizes. From our problem, we take the total voted from both groups (\(45 + 56 = 101\)) and divide by the sum of individuals (\(70 + 100 = 170\)). This gives us a pooled proportion of \( \frac{101}{170} = 0.594 \).

Pooled proportion is crucial because it serves as a single estimate of the common proportion under the assumption that the true proportions in the populations being compared are the same.

Normal Distribution Application

In the context of hypothesis testing for two proportions, we apply the normal distribution to approximate the sampling distribution of the difference between the proportions. This approximation holds true under certain conditions, such as a large enough sample size from each group. In the exercise, calculating the test statistic assumes the distribution of the test statistic under the null hypothesis follows a normal distribution.

The significance of understanding this application lies in the ability to quantify the likelihood of differences observed in sample proportions occurring by random chance. By comparing our test statistic to critical values from the normal distribution, we can make informed decisions about the validity of our null hypothesis.

Null and Alternative Hypotheses Formulation

The null hypothesis (\(H_0\)) for two proportions typically states that there is no difference between the population proportions (\(p1 = p2\)), while the alternative hypothesis (\(H_a\)) asserts that there is a difference (\(p1 eq p2\)). The entire hypothesis test is designed around the possibility of rejecting the null hypothesis in favor of the alternative.

The formulation of these hypotheses is a pivotal step in hypothesis testing because they create a basis for comparison and decision making. Making them explicit also clarifies what evidence is needed to support a claim of a genuine difference between groups, as seen in the voting proportion example given.

Test Statistic Calculation

The test statistic is calculated as a standardized value that compares the observed difference to the expected distribution under the null hypothesis. For our two proportion hypothesis test, it is given by the formula \( Z = \frac {(p1 - p2) - 0 }{ \sqrt {P(1 - P)(\frac{1}{n1} + \frac{1}{n2} )}} \), where \(p1\) and \(p2\) are the sample proportions, \(P\) the pooled proportion, and \(n1\) and \(n2\) the sample sizes. This formula centralizes and scales the difference in proportions, allowing us to compare it against the standard normal distribution.

The concept of a test statistic is significant because it forms the bridge between the data collected and the probability model applied. It quantifies the evidence against the null hypothesis and determines the outcome of the hypothesis test.