In a test of the quality of two television commercials, each...

In a test of the quality of two television commercials, each commercial was shown in a separate test area six times over a one-week period. The following week a telephone survey was conducted to identify individuals who had seen the commercials. Those individuals Number Who Saw Commercial Number Who Recalled Message $$ \begin{array}{cc}{\text { Commercial A }} & {\text { Commercial } \mathbf{B}} \\ {150} & {200} \\ {63} & {60}\end{array} $$ a. Use $\alpha=.05$ and test the hypothesis that there is nodifference in the recall proportions for the two commercials. b. Compute a 95$\%$ confidence interval for the difference between the recall proportions for the two populations.

In a test of the quality of two television commercials, each commercial was shown in a
separate test area six times over a one-week period. The following week a telephone survey
was conducted to identify individuals who had seen the commercials. Those individuals
Number Who Saw Commercial
Number Who Recalled Message $$ \begin{array}{cc}{\text { Commercial A }} & {\text { Commercial } \mathbf{B}} \\ {150} & {200} \\ {63} & {60}\end{array} $$
a. Use $\alpha=.05$ and test the hypothesis that there is nodifference in the recall proportions for the two commercials.
b. Compute a 95$\%$ confidence interval for the difference between the recall proportions for the two populations.

Key Concepts

Hypothesis Testing for Two Proportions

This concept involves comparing two independent sample proportions to determine if there is a statistically significant difference between two population proportions. The process requires setting up a null hypothesis that typically states there is no difference between the proportions and an alternative hypothesis suggesting otherwise. This test often uses a z-test when the sample sizes are large enough for the normal approximation to be valid.

Standard Error of the Difference Between Proportions

This concept refers to the measure of variability in the difference between two sample proportions. It is calculated by combining the variances of the individual proportions and is crucial for accurately estimating the uncertainty in the comparison of the two groups. The standard error is a key component for both hypothesis testing and constructing confidence intervals.

Confidence Interval for the Difference in Proportions

A confidence interval provides a range of plausible values for the true difference between two population proportions. It is constructed using the observed difference between the sample proportions and the corresponding standard error. This interval offers a quantitative measure of uncertainty and helps in understanding the magnitude and precision of the estimated difference.

Significance Level and p-value

The significance level (alpha) is a predetermined threshold used to decide whether to reject the null hypothesis in hypothesis testing, often set at 0.05. The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. Comparing the p-value to the significance level helps in determining the statistical significance of the observed difference.

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