A Quinnipiac poll conducted on February 20 , 2018 , found...

A Quinnipiac poll conducted on February 20 , 2018 , found that 824 people out of 1249 surveyed favored stricter gun control laws. A survey conducted one week later on February 28 , 2018 , by National Public Radio found that 754 out of 1005 people surveyed favored stricter gun control laws. a. Find both sample proportions and compare them. b. Test the hypothesis that the population proportions are not equal at the $0.05$ significance level. c. After conducting the hypothesis test, a further question one might ask is what is the difference between the two population proportions? Find a $95 \%$ confidence interval for the difference between the two proportions and interpret it. How does the confidence interval support the hypothesis test conclusion?

A Quinnipiac poll conducted on February 20 , 2018 , found that 824 people out of 1249 surveyed favored stricter gun control laws. A survey conducted one week later on February 28 , 2018 , by National Public Radio found that 754 out of 1005 people surveyed favored stricter gun control laws.
a. Find both sample proportions and compare them.
b. Test the hypothesis that the population proportions are not equal at the $0.05$ significance level.
c. After conducting the hypothesis test, a further question one might ask is what is the difference between the two population proportions? Find a $95 \%$ confidence interval for the difference between the two proportions and interpret it. How does the confidence interval support the hypothesis test conclusion?

Key Concepts

Sample Proportion

The sample proportion is a statistic that represents the fraction of individuals in a sample who exhibit a particular characteristic. It is computed by dividing the number of individuals with the trait by the total number surveyed and serves as a point estimate of the population proportion. This concept is fundamental in inferential statistics, as it provides the basis for further analysis and hypothesis testing regarding population parameters.

Hypothesis Testing for Two Proportions

This concept involves comparing the proportions from two independent samples to determine whether there is a statistically significant difference between the respective populations. The process typically involves formulating null and alternative hypotheses, calculating a test statistic (often using a z-test), and using a significance level to decide whether to reject the null hypothesis. It is a key tool in assessing differences in categorical outcomes between groups.

Standard Error and the Z-Test

The standard error measures the variability or dispersion of the difference between two sample proportions assuming a null hypothesis of no difference. It is used to standardize the difference and compute a z-test statistic. The z-test is a common method in hypothesis testing for two proportions, allowing researchers to determine how many standard errors the observed difference is away from the hypothesized difference, thereby facilitating the calculation of p-values.

Confidence Interval for the Difference of Proportions

A confidence interval for the difference between two population proportions provides a range of plausible values for the true difference based on sample data. Constructed at a specified confidence level (such as 95%), it not only estimates the magnitude of the difference but also indicates the reliability and precision of the estimate. When the interval does not contain zero, it supports the conclusion that there is a statistically significant difference between the two proportions, aligning with the findings from a hypothesis test.

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