Question

3. Testing for equal proportions

Imagine that you are contracted by a local news provider to study consumer demographics in relation to three different types of news media: print (newspaper), Internet, and television. In prior market research, the company has classified each of its customers as receiving news content primarily from only one of these three sources, and as either urban or rural residents. In order to help design effective marketing strategies, you are asked to perform a test for equality of proportions to determine whether there is a significant difference in the proportion of consumers who live in urban versus rural areas for the three media types that are offered.

The three population proportions that you are interested in are:

	p₁ = proportion of urban consumers for the population of newspaper readers
	p₂ = proportion of urban consumers for the population of Internet news readers
	p₃ = proportion of urban consumers for the population of TV news consumers

You conduct a hypothesis test with a 0.05 level of significance to determine whether the proportion of urban consumers is the same for all three news sources. The null and alternate hypotheses for your test are:

H₀:

Ha: You collect a random sample of 1,119 consumers of the company’s news content. You find that 212 of the 299 newspaper consumers, 315 of the 379 Internet consumers, and 245 of the 441 TV consumers lived in urban areas. The data are summarized in the following table: 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 each population, assuming H₀ is true (round the frequencies to the nearest whole number). (Note: Due to rounding, the row and column totals for your version of this problem may not match the values shown 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-square distribution with____ degrees of freedom. The chi-square test statistic for your test is χ² =   . Use the following table of selected values of the chi-square distribution 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 null hypothesis. You   that there is a difference in consumer demographics among the three news media sources.

Answer 1

NOTE- The values under the bracket are Expected frequencies.

Expected frequencies calculated as followed:

Expected frequencies= Row total* Column total / Grand total

And,

X²= 73.053

degrees of freedom = (r-1)(c-1)= (2-1)(3-1)= 2

As the critical value is 5.991

X²_calculated > X²_tabulated, which means we would reject the null hypothesis.

There is a difference in consumer demographics among the three news media sources.