Question

Create a case (please refer to the sample below) on the application of inferences about the difference between two population means (σ1 and σ2 unknown) and explain the hypothesis tests until conclusion.

EXAMPLE:

22% express an interest in seeing XYZ television show. KL Broadcasting Company ran commercials for this XYZ television show and conducted a survey afterwards. 1532 viewers who saw the commercials were sampled and 414 said that they would watch XYZ television show.

What is the point estimate of the proportion of the audience that said they would watch the television show after seeing the commercials?

At α=0.05, determine whether the intent to watch the television show significantly increased after seeing the television commercials. Formulate the appropriate hypothesis, compute the p-value, and state your conclusion.

Answer 1

The point estimate of the proportion of the audience that said they would watch the television show after seeing the commercials is the sample proportion given by 414/1532 = 0.2702

Here we would like to test the hypothesis:

To test the above hypothesis we'll be using the one-sample Z test for proportions. The test statistic is given by:

. The p-value of the test is:

P(Z>4.7465)=1.03*10^-6. Which is very less than the significance level α=0.05. Thus we reject the null hypothesis.

Hence we conclude that there is a significant increase in the intent to watch the television after seeing the television commercials

A test case:

Forced Vital Capacity (FVC) is the volume (in milliliters) of air that an individual can exhale in 6 seconds. FVC was obtained for a sample of children not exposed to parental smoking and for a sample of children exposed to parental smoking. Is the mean FVC lower in the population of children exposed to parental smoking? Use α=0.05. Here is the table:

Here we're testing

Here we'll be using a two-sample t-test with unequal variance. the test statistic is given by:

and the degrees of freedom is given by:

Thus we can the degrees of freedom to  48. Thus the critical value at a significance level α=0.05 is T_0.95,48=-1.677. The test statistic is less than the critical value hence we reject the null hypothesis. We conclude that Lung capacity is significantly impaired in children exposed to parental smoking, compared with children not exposed to parental smoking.