A 30 -minute consumer survey mailed to 500 adults aged 25-34...

A 30 -minute consumer survey mailed to 500 adults aged $25-34$ included a $$\$ 10$$ gift certificate to Borders. The same survey was mailed to 500 adults aged $25-34$ without the gift certificate. There were 185 responses from the first group and 45 from the second group. (a) At $\alpha=.025$, did the gift certificate increase the response rate by at least 20 percent? (b) Find the $p$-value.

Key Concepts

Sampling Distribution of the Difference in Proportions

Understanding the sampling distribution of the difference in two independent proportions is key in hypothesis testing. It allows for the estimation of the standard error, which quantifies the variability of the sample difference under repeated sampling. This distribution, when approximately normal, forms the basis of the z-test and is used to compute the test statistic, thereby enabling researchers to make inferences about the population parameters.

Significance Level and p-Value

The significance level (?) is the threshold for deciding whether an observed effect is statistically significant. The p-value is the probability of observing an effect as extreme as, or more extreme than, the effect in your sample data under the assumption that the null hypothesis is true. If the p-value is smaller than the significance level, the null hypothesis is rejected. These concepts are central to determining the strength of evidence against the null hypothesis in hypothesis testing.

Two-Proportion Z-Test

The two-proportion z-test is a statistical method used to compare the proportions from two independent samples. It assesses whether there is a significant difference between the two sample proportions, typically by calculating the difference, standard error, and then using a z-statistic. This test is commonly applied when analyzing survey responses or experiments where outcomes are dichotomous (e.g., response vs. no response).

Hypotheses Formulation

In any hypothesis test, it is crucial to clearly define the null and alternative hypotheses. For a test comparing two proportions, the null hypothesis usually states that the difference between the proportions is equal to a specified value (often zero or another benchmark value), while the alternative hypothesis asserts that the difference is greater than, less than, or not equal to that benchmark. The precise formulation guides the direction of the test and the interpretation of results.

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