Question

Quebec vs Texas Secession In Example 6.4 on page 408 we analyzed a poll of 800 Quebecers, in which \(28 \%\) thought that the province of Quebec should separate from Canada. Another poll of 500 Texans found that \(18 \%\) thought that the state of Texas should separate from the United States. \({ }^{38}\) (a) In the sample of 800 people, about how many Quebecers thought Quebec should separate from Canada? In the sample of 500 , how many Texans thought Texas should separate from the US? (b) In these two samples, what is the pooled proportion of Texans and Quebecers who want to separate? (c) Can we conclude that the two population proportions differ? Use a two- tailed test and interpret the result.

Short answer

About 224 Quebecers and 90 Texans want their regions to secede. The pooled proportion is approximately 0.2415 or 24.15%. The z-score is approximately 3.82, which is greater than 3, suggesting that the population proportions do differ significantly.

Step-by-step solution

Calculation of Sample Proportions

First, calculate the number of people who want their respective regions to secede. In Quebec, \(28\%\) of 800 people thought Quebec should separate from Canada. This can be calculated as \((28/100)*800 \approx 224\). Similarly, in Texas, \(18\%\) of 500 people thought Texas should separate from the US. This can be calculated as \((18/100)*500 \approx 90\). The calculated numbers are the estimates of people in both samples who want their regions to secede.

Pooled Proportion Calculation

To calculate the pooled proportion, the total number of successes from both samples is computed and then divided by the total size of both samples. The total number of people who want their regions to secede is \(224+90 = 314\). The total sample size is \(800 + 500 = 1300\). Hence the pooled proportion is \(314/1300 \approx 0.2415\).

Two-tailed Test Execution and Interpretation

The two-tailed test checks if the proportions in both samples are significantly different. Find the standard error by using the formula \(\sqrt{p(1-p)(1/n1+1/n2)}\) where p is the pooled proportion and n1 and n2 are the sample sizes. Substituting the given values, the standard error is \(\sqrt{0.2415(1-0.2415)(1/800+1/500)} \approx 0.0262\). Find the z-score by subtracting the proportions and dividing by the standard error. This gives \(|0.28-0.18|/0.0262 \approx 3.82\). Generally, a z-score above 3 or below -3 indicates that the proportions are significantly different. Hence, it can be concluded that the population proportions do differ significantly.

Key concepts

Sample Proportions Calculation

Understanding the sample proportion calculation is crucial in statistics, especially when analyzing poll results or survey data. A sample proportion represents a percentage of participants in a study that have a particular characteristic. For instance, let's consider a scenario where we want to find out how many people support a particular policy within a group.

To calculate the sample proportion, you would follow a simple formula: \[ \text{Sample Proportion} = \frac{\text{Number of people with the characteristic}}{\text{Total number of people in the sample}} \]
This can also be expressed as a percentage. In the case of the Quebecers who support secession, with a sample size of 800 and 28% supporting secession, the number of supporters is calculated by multiplying 28% (or 0.28) by 800, resulting in approximately 224 individuals.

Pooled Proportion

When dealing with two or more samples, the pooled proportion is a tool that combines the information from these samples to give an overall proportion. This is particularly helpful when we wish to compare two groups and see if there is a significant difference regarding a specific characteristic.

The formula used to calculate the pooled proportion is:\[ \text{Pooled Proportion} = \frac{\text{Total number of successes in all samples}}{\text{Total sample size across all samples}} \]
In the context of the Quebec and Texas secession polls, we add the number of individuals supporting secession in Quebec (224) with those in Texas (90) to get a total number of 'successes.' Then, we divide this total by the combined sample size (1300). As a result, we obtain an overall proportion of the study population who favor secession, allowing us to conduct further comparative analyses.

Two-tailed Test

A two-tailed test in statistics is used when we're interested in determining whether there is a difference between two groups in either direction. This type of test is applied when the direction of the difference isn't specified beforehand. In the context of comparing proportions from two different populations, such as Quebecers and Texans on the issue of secession, a two-tailed test can assess whether the observed difference in sample proportions is statistically significant.

The key steps in this test involve calculating the standard error and then determining the z-score. The z-score measures the number of standard errors the observed difference is away from zero (no difference). If the z-score falls beyond the threshold set for significance (often at z-scores greater than 3 or less than -3), we can conclude that there is a significant difference between the two population proportions. The z-score in our example of secession preferences is approximately 3.82, indicating that the difference in secession support between the two populations is indeed significant.