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 were asked to state the primary message in the commercials. The following results were recorded. $\begin{array}{lcc} & \text { Commercial A } & \text { Commercial B } \\ \text { Number Who Saw Commercial } & 150 & 200 \\ \text { Number Who Recalled Message } & 63 & 60\end{array}$ a. Use $\alpha=.05$ and test the hypothesis that there is no difference 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 were asked to state the primary message in the commercials. The following results were recorded. $\begin{array}{lcc} & \text { Commercial A } & \text { Commercial B } \\ \text { Number Who Saw Commercial } & 150 & 200 \\ \text { Number Who Recalled Message } & 63 & 60\end{array}$
a. Use $\alpha=.05$ and test the hypothesis that there is no difference 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

Confidence Interval for the Difference

A confidence interval for the difference between two proportions provides a range of plausible values for the true difference in the populations' proportions. A 95% confidence interval, for example, offers a range within which the actual difference is expected to lie with 95% certainty, thereby quantifying the magnitude and direction of the difference rather than just testing for its presence.

Sampling Distribution

The sampling distribution refers to the probability distribution of a given statistic, such as the difference between two sample proportions, if the same sampling process were repeated many times. For large samples, the central limit theorem assures that this distribution approximates a normal distribution, which is a key assumption when using Z-tests and constructing confidence intervals for comparing proportions.

Significance Level (?)

The significance level, denoted by ?, represents the threshold for rejecting the null hypothesis. It is the probability of making a Type I error—rejecting the null hypothesis when it is actually true. In this problem, an ? of 0.05 implies that there is a 5% risk of incorrectly concluding a difference in recall proportions between the commercials when no difference exists.

Standard Error and Test Statistic

The standard error measures the variability of the difference between the two sample proportions and is derived from the pooled proportion when assuming the null hypothesis is true. By dividing the observed difference between the proportions by this standard error, a test statistic (usually a Z-score) is obtained, which is then compared to critical values from the normal distribution to determine statistical significance.

Two-Sample Proportion Test

A two-sample proportion test compares the proportions from two independent groups to determine if there is a statistically significant difference between them. In this case, the test evaluates whether the proportion of individuals who recalled the message differs between the two television commercials.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. In this context, it involves setting up a null hypothesis that there is no difference between the recall proportions of the two commercials and an alternative hypothesis that there is a difference, then using sample data to determine which hypothesis is more consistent with the observed outcomes.

Pooled Proportion

When testing the equality of two proportions under the null hypothesis, the pooled proportion is used as a combined estimate of the common success probability. This combined measure is calculated by summing the successes from both groups and dividing by the total number of observations, and it is essential for calculating the standard error in the hypothesis test.

Transcript

0:00 We have two commercials.

00:01 We have commercial a and we have commercial b.

00:05 And commercial a, we had 63 out of 150 people who saw the commercial, recalled the details.

00:14 So now i'll call that proportion, p hat for a.

00:17 63 out 150 people recalled the message of a commercial after a week of having had seen the commercial.

00:27 And those for b, there was a sample size of 200, of which 60 of those people recalled the message for commercial b.

00:38 And we want to know, do we have evidence that there is, whether there is a difference between these two proportions? so we would be assuming that the proportion of a is equal to the proportion of b.

00:53 And alternately, we just want to find if there's a difference.

00:55 So we would have it be a, whoops, that should be a, not equal to.

01:02 So we're doing a two -tailed test.

01:05 And let's just get the decimal.

01:06 I know what this decimal is.

01:07 This is 0 .3, dividing top and bottom by 2, but 63 divided by 150.

01:13 Let's see what that is.

01:14 Helps to turn the calculator on my goodness.

01:17 That is 42%.

01:18 All right.

01:20 So we want to find, do we have evidence at a 5 % significance level, whether there's a difference here.

01:27 So picture -wise, think what we're doing.

01:30 We are assuming that the difference between the two proportions, i'll say pa minus pb, we're assuming it's equal to zero.

01:39 Now, we can definitely see that the difference is not zero.

01:43 We're getting a difference of like 12%.

01:46 And because we're doing a two -tail test, this difference is considered just as likely as symmetrically on the other side, a difference the opposite way.

01:57 So that's what we want to find.

01:59 We want to find what these two areas are together to be our p value.

02:05 Okay, and we need to find our test statistics.

02:08 So our test statistic is going to be a z, and we're going to take the difference between the two, so 0 .42 minus 0 .3, and then we have to find the pooled p, and i'm just going to call that good old p half.

02:21 And the pooled p is 123.

02:26 Over and the sum of these two is 350.

02:31 So our test statistic is going to be the pooled p and then one minus the pooled p, which looks like that's going to be 227 over 350.

02:48 So again, if i double check, 350 minus the 123.

02:52 Yep, that's 227 over that.

02:54 So pooled p, 1 minus pooled p, and then times one over the sample size plus one over the sample size.

03:02 And that will be our test statistic...