With the proliferation of the Internet and 24-hour cable news outlets, it has become much easier for people to hear much more information, much more quickly. This has raised concerns that some news organizations may try to report information before it has been properly verified. A media believability survey has been conducted since 1985 (under the direction of the Pew Research Center for the People and the Press since 1996) to examine whether different news organizations have been losing credibility over time.
The survey is based on telephone interviews among a national sample of adults 18 years or older living in the continental United States. One question asked respondents to rate believability on a 4-to-1 scale, where 4 means they believe all or most of what the organization says and 1 means they believe almost nothing.
Rotating the list helps prevent order effects. Without rotation, organizations listed earlier could be over-selected or rated differently simply because of position in the interview sequence.
Including cell phones reduces coverage bias. If only landlines are sampled, important groups (especially younger adults and cell-only households) are underrepresented.
Convert these percentages into observed counts among those who felt they could rate their daily newspaper. In other words, eliminate the "can’t rate" folks from consideration.
About 92.9% could rate the paper, so the count of raters is approximately \(1004(0.929) \approx 933\text{.}\) Converting the category percentages among raters gives counts of about 181 (rating 4), 371 (rating 3), 261 (rating 2), and 120 (rating 1).
Here the response variable is categorical with four ordered levels (an ordinal scale). In Investigation 5.1, the response variable was binary categorical (female/male).
A similar study was also done in 1998 (922 respondents able to rate) and in 2012 (922 able to rate). A corresponding two-way table of counts across the three years is shown below.
The percentage declines from about \(265/922 \approx 28.7\%\) in 1998 to \(181/933 \approx 19.4\%\) in 2006, then is about \(183/922 \approx 19.8\%\) in 2012. So there is a large drop from 1998 to 2006 and little change afterward.
If the proportion who believe all or almost all of what they read was the same in all three populations, how many in the 2006 sample would you expect to believe all or almost all of what they read?
State the null and alternative hypotheses for assessing whether the sample data provide strong evidence that the distribution of responses differs among the three years.
where \(r\) is the number of rows and \(c\) is the number of columns in the two-way table. This statistic often follows a chi-squared distribution with degrees of freedom\((r-1)(c-1)\text{.}\)
The data must be in “stacked format” (one column with the EV categories, one column with the RV categories). Each row can represent an observation or each row can represent a unique combination of variables and then you can a column of counts (“tallied”).
A typical output gives \(\chi^2 \approx 45.22\) with \(df=(4-1)(3-1)=6\) and a very small p-value (\(p < 0.0001\)). We reject the null hypothesis and conclude that the believability distribution differs by year.
Use technology to carry out a chi-squared test to decide whether the difference in the population proportions giving a largely believable rating differed significantly between the two years. Report the standardized statistic, degrees of freedom, and p-value.
The chi-square test gives about \(\chi^2 \approx 19.9\) with \(df=1\) and \(p < 0.0001\text{,}\) indicating a statistically significant difference between years.
Now carry out a two-sided two-proportion \(z\)-test for this table. Report the standardized statistic and p-value. How do the p-values compare? What do you think is the relationship between the standardized statistics?
The two-proportion test gives about \(z \approx -4.46\) with a two-sided \(p < 0.0001\text{.}\) The p-values agree closely with the chi-square test, and the statistics satisfy \(z^2 \approx \chi^2\text{.}\)
The chi-squared procedure can be used to compare two or more proportions. When there are only two proportions, the procedure is equivalent to a two-sided \(z\)-test for proportions from Chapter 3. The chi-squared test statistic is equal to \(z^2\) and the chi-squared p-value is equal to the two-sided p-value for the two-sample \(z\)-test. If the alternative hypothesis is two-sided, you can use either procedure. If the alternative hypothesis is one-sided, then you should carry out the two-sample \(z\)-test to obtain the one-sided p-value. If there are more than two proportions, then you must use the chi-squared procedure, which will only assess whether or not at least one population proportion differs from the others.
For a \(2\times 2\) two-way table, another alternative is Fisher’s Exact Test as you learned in Chapter 3. It is always appropriate to carry out Fisher’s Exact Test (if you are willing to fix the row and column totals), though it may be a bit less convenient with huge sample sizes (which becomes less of an issue with each new computer chip).
However, of all three of these procedures, the two-sample \(z\)-procedure is the only one that also enables us to calculate a confidence interval to estimate the magnitude of the difference in the two population proportions.
Keep in mind that chi-squared tests look for evidence of any association. They do not care which is the explanatory variable and which is the response variable, and they do not look for specific types of association, like time trend.
In February of 1993, NBC News admitted that it staged the explosion of a General Motors truck during a segment of the program Dateline NBC in November of 1992. The segment included crash footage that explosively showed how the gas tanks of certain old GM trucks could catch fire in a sideways collision. In a nationwide poll of adults (Times Mirror News Interest Index) conducted in August, 1989, 1507 respondents gave NBC news the following believability ratings.
