Question

Problem 1 The director of Gainesville Sun newspaper is studying the relationship between the type of com- munity in which a subscriber resides and the section of the newspaper he or she reads first. For a sample of readers, she collected the sample information in the following table: City Suburb Rural National News Sports 170 124 120 130 130 9 0 Comics 90 100 88 a. At the 95% confidence level, can we conclude that there is a relationship between the type of community where the person resides and the section of the paper read first, i.e. are they dependent? b. What is the largest chi-square value? How would you interpret this? c. Is the number of people that read national news in rural areas greater or less than expected? d. Is the number of people that read comics in cities greater or less than expected?

Answer 1

The contingency table is given as follows:

Observed frequency
	National News	Sports	Comics	Total
City	32	30	28	90
Suburb	74	64	42	180
Rural	110	25	15	150
Total	216	119	85	420

Null Hypothesis: Ho – the type of community and section of newspaper are independent

Alternative hypothesis: H1 – the type of community and section of newspaper are dependent

Determining test statistics  

χ2test = Σ[(f0 – fe)2/fe]

Where,

f0 = observed frequency of the category

fe = expected frequency of the category

The expected frequency of the category is calculated as follows:

fe = (∑rf0)(∑cf0)/(n)

∑rf0 = total observed frequency of respective row

∑cf0 = total observed frequency of respective column

n = total number of observation = 420

Expected frequency for city resident and national news subscriber category = (90)(216)/(420) = 46.29

Similarly expected frequencies for the remaining category are as follows:

Expected frequency
	National News	Sports	Comics	Total
City	46.29	25.50	18.21	90
Suburb	92.57	51.00	36.43	180
Rural	77.14	42.50	30.36	150
Total	216	119	85	420

Contribution of category to Chi-square test is calculated as follows:

Contribution = (f0 – fe)2/fe

For City and national news category the contribution is calculated as follows:

Contribution = (32 – 46.29)2/(46.29) = 4.41

Similarly, contributions of other categories are obtained as follows:

Contribution
	National News	Sports	Comics	Total
City	4.41	0.79	5.26	10.46
Suburb	3.73	3.31	0.85	7.89
Rural	13.99	7.21	7.77	28.97
Total	22.13	11.31	13.88	47.32

test statistics  

χ2test = Σ[(f0 – fe)2/fe]

Value of test statistics = χ2test = 47.32

Critical value

Degree of freedom = (r – 1)(c – 1) = (3 – 1)(3 – 1) = 4

Significance level = 1 – 0.95 = 0.05

From table of Chi-square, χ2critical = 9.488

Using the critical value approach, the decision rule is to reject the null hypothesis if χ2test >= χ2critical

Since, χ2test (47.32) >= χ2critical (9.488), we reject null hypothesis that the type of community and the section of the newspaper subscriber reads are independent at a = 0.05.

From table, χ2test = 60.44 and degree of freedom = 4, p-value < 0.005

Since p-value < a (0.05), reject null hypothesis.

Thus, there is sufficient data to conclude that there is relationship between the type of community and the section of the newspaper subscriber reads at 95% confidence level.