Question

A television station wishes to study the relationship between viewership of its 11 p.m. news program and viewer age (18 years or less, 19 to 35, 36 to 54, 55 or older). A sample of 250 television viewers in each age group is randomly selected, and the number who watch the station’s 11 p.m. news is found for each sample. The results are given in the table below.

		Age Group
Watch 11 p.m. News?	18 or less	19 to 35	36 to 54	55 or Older	Total
Yes	49	52	67	79	247
No	201	198	183	171	753
Total	250	250	250	250	1,000

(a) Let p₁, p₂, p₃, and p₄ be the proportions of all viewers in each age group who watch the station’s 11 p.m. news. If these proportions are equal, then whether a viewer watches the station’s 11 p.m. news is independent of the viewer’s age group. Therefore, we can test the null hypothesis H₀ that p₁, p₂, p₃, and p₄ are equal by carrying out a chi-square test for independence. Perform this test by setting α = .05. (Round your answer to 3 decimal places.)

χ2

=            

so

H₀: independence

(b) Compute a 95 percent confidence interval for the difference between p₁ and p₄. (Round your answers to 3 decimal places. Negative amounts should be indicated by a minus sign.)

95% CI: [ , ]

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: p₁= p₂ = p₃ = p₄

Alternative hypothesis: At least one of the null hypothesis statements is false.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square test for homogeneity.

Analyze sample data. Applying the chi-square test for homogeneity to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = (r - 1) * (c - 1) = (4 - 1) * (2- 1)
D.F = 3
E_r,c = (n_r * n_c) / n

Χ² = 12.53

where DF is the degrees of freedom, r is the number of populations, c is the number of levels of the categorical variable, n_r is the number of observations from population r, n_c is the number of observations from level c of the categorical variable, n is the number of observations in the sample, E_r,c is the expected frequency count in population r for level c, and O_r,c is the observed frequency count in population r for level c.The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 12.53

We use the Chi-Square Distribution Calculator to find P(Χ² > 12.53) = 0.006

Interpret results. Since the P-value (0.006) is less than the significance level (0.05), we reject the null hypothesis.

b) 95 percent confidence interval for the difference between p₁ and p₄ is C.I = (- 0.196, - 0.044).

C.I = (0.196 - 0.316) + 1.96*0.03866

C.I = - 0.12 + 0.07577

C.I = (- 0.196, - 0.044)