3. Testing for equal proportions
Imagine that you are contracted by a local news provider tostudy consumer demographics in relation to three different types ofnews media: print (newspaper), Internet, and television. In priormarket research, the company has classified each of its customersas receiving news content primarily from only one of these threesources, and as either urban or rural residents. In order to helpdesign effective marketing strategies, you are asked to perform atest for equality of proportions to determine whether there is asignificant difference in the proportion of consumers who live inurban versus rural areas for the three media types that areoffered.
The three population proportions that you are interested inare:
| p? = proportion of urban consumers for the population ofnewspaper readers |
| p? = proportion of urban consumers for the population ofInternet news readers |
| p? = proportion of urban consumers for the population of TVnews consumers |
You conduct a hypothesis test with a 0.05 level of significanceto determine whether the proportion of urban consumers is the samefor all three news sources. The null and alternate hypotheses foryour test are:
| H?: |
| Ha: You collect a random sample of 1,119 consumers of the company’snews content. You find that 212 of the 299 newspaper consumers, 315of the 379 Internet consumers, and 245 of the 441 TV consumerslived in urban areas. The data are summarized in the followingtable: Sample Results | | News Source |
|---|
| | Newspaper | Internet | TV | Total | | Consumer | Urban | 212 | 315 | 245 | 772 | | Rural | 87 | 64 | 196 | 347 | | Total | 299 | 379 | 441 | 1,119 |
Complete the following table of expected frequencies for eachpopulation, assuming H? is true (round the frequencies to thenearest whole number). (Note: Due to rounding, the row and columntotals for your version of this problem may not match the valuesshown in the table.) Expected Frequencies | | News Source |
|---|
| | Newspaper | Internet | TV | Total | | Consumer | Urban | | | | 772 | | Rural | | | | 347 | | Total | 299 | 379 | 441 | 1,119 |
To conduct your hypothesis test, you use a chi-squaredistribution with____ degrees of freedom. The chi-square teststatistic for your test is ?² = . Use the following table of selected values of the chi-squaredistribution to reach a conclusion about your null hypothesis: Degrees of Freedom | Area in Upper Tail |
|---|
.10 | .05 | .025 | .01 |
|---|
| 1 | 2.706 | 3.841 | 5.024 | 6.635 | | 2 | 4.605 | 5.991 | 7.378 | 9.210 | | 3 | 6.251 | 7.815 | 9.348 | 11.345 | | 4 | 7.779 | 9.488 | 11.143 | 13.277 | | 5 | 9.236 | 11.070 | 12.833 | 15.086 | | 6 | 10.645 | 12.592 | 14.449 | 16.812 | | 7 | 12.017 | 14.067 | 16.013 | 18.475 | | 8 | 13.362 | 15.507 | 17.535 | 20.090 | | 9 | 14.684 | 16.919 | 19.023 | 21.666 | | 10 | 15.987 | 18.307 | 20.483 | 23.209 |
With a 0.05 level of significance, you the nullhypothesis. You that there is a difference in consumerdemographics among the three news media sources. |
