Question

Gonzaga University professors conducted a study of television commercials and published their results in the Journal of Sociology, Social Work and Social Welfare (Vol. 2, 2008). The key research question was as follows: “Do television advertisers use religious symbolism to sell goods and services?” In a sample of 797 TV commercials collected ten years earlier, only 16 commercials used religious symbolism. Of the sample of 1,499 TV commercials examined in the more recent study, 51 commercials used religious symbolism. Conduct an analysis to determine if the percentage of TV commercials that use religious symbolism has changed over time. If you detect a change, estimate the magnitude of the difference and attach a measure of reliability to the estimate.

Short answer

We have sufficient evidence to conclude that the percentage of TV commercials that use religious symbolism has changed over time.

Step-by-step solution

Given information

For the sample collected ten years earlier,

The size of the sample is $n_1=797$.

The proportion of commercials that use religious symbolism is

$\begin{array}{rcl}{\hat{p}}_1& =& \frac{x_1}{n_1}\\ & =& \frac{16}{797}\\ & =& 0.0201\end{array}$

Also, for the sample collected more recently

The size of the sample is $n_2=1499$.

The proportion of commercials that use religious symbolism is

$$\begin{gathered} {\hat{p}}_2=\frac{x_2}{n_2} \\ =\frac{51}{1499} \\ =0.0340 \end{gathered}$$

Hypothesis Testing

Hypothesis testing is the process of testing the statements made by the researcher or any other concerned person using the probability sampling method's sample data.

Setting up the Hypotheses

We have to test whether the percentage of TV commercials that use religious symbolism has changed over time or not.

Let

$p_1:$The population proportion of commercials used religious symbolism ten years earlier.

$p_2:$The population proportion of commercials recently used religious symbolism.

The null and alternative hypotheses are

$H_0:p_1=p_2$

The percentage of TV commercials that use religious symbolism has not changed.

Against

$H_a:p_1\ne p_2$

The percentage of TV commercials that use religious symbolism has changed over time.

Calculating the test statistic

The test statistic for testing these hypotheses is

$$\begin{gathered} Z=\frac{{\hat{p}}_1-{\hat{p}}_2}{\sqrt{\frac{{\hat{p}}_1\left(1-{\hat{p}}_1\right)}{n_1}+\frac{{\hat{p}}_2\left(1-{\hat{p}}_2\right)}{n_2}}} \\ =\frac{0.0201-0.0340}{\sqrt{\frac{0.0201\left(1-0.0201\right)}{797}+\frac{0.0340\left(1-0.0340\right)}{1499}}} \\ =\frac{-0.0139}{\sqrt{0.000024713+0.000021911}} \\ =\frac{-0.0139}{0.00683} \\ =-2.04 \end{gathered}$$

Calculating the p-value

The p-value for the obtained test statistic is

$$\begin{gathered} p-value=2\times P\left(Z>\left|z\right|\right) \\ =2\times P\left(Z>2.04\right) \\ =2\times \left(1-P\left(Z⩽2.04\right)\right) \\ =2\times \left(1-0.9793\right) \\ =2\times 0.0207 \\ =0.0414 \end{gathered}$$

Decision for the null hypothesis

We can see that

$p-value<0.05$

Hence, we reject the null hypothesis $H_0$.

Conclusion

At a 5% significance level, we have sufficient evidence to conclude that the percentage of TV commercials that use religious symbolism has changed over time.

The magnitude of the difference

The magnitude of the difference is given as

$\begin{array}{rcl}d& =& \left|{\hat{p}}_1-{\hat{p}}_2\right|\\ & =& \left|0.0201-0.0340\right|\\ & =& 0.0139\end{array}$

The measure of reliability cannot be computed here, as the true difference between the population proportions is unknown.