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MA 220
Exercise 6 - Chi-square test
Contingency table
Estimated cell count: Eˆij =
ricj n
, where ri: the total for row i and cj : the total for column
j.
Test statistic: X^2 = Σ
(Oij − Eˆij )^2 E^ ˆij
, Oij : observed cell count
df = (r − 1)(c − 1), where r is the number of rows and c is the number of columns
- A survey of 400 respondents produced these cell counts in a 2 × 3 contingency table:
Columns
Rows 1 2 3 Total
1 37 34 93 164
2 66 57 113 236
Total 103 91 206 400
(a) If you wish to test the null hypothesis of ”independence” - that the probability that a response falls in any one row is independent of column it falls in - and you plan to use a chi-square test, how many degrees of freedom will be associated with the χ^2 statistic?
(b) Find the value of the test statistic.
(c) Find the rejection region for α =. 01
(d) Conduct the test and state your conclusions.
(e) Find the approximate p− value for the test and interpret its value.
MA 220
Exercise 6 - Chi-square test
- A study to determine the effectiveness of a drug (serum) for arthritis resulted in the comparison of two groups, each consisting of 200 arthritic patients. One group was inoculated with serum; the other received a placebo (an inoculation that appears to contain serum but actually is nonactive). After a period of time, each person in the study was asked to state whether his or her arthritic condition had improved. These are the results: You want to know whether these data present sufficient evidence to indicate
Treated Untreated
Improved 117 74
Not Improved 83 126
that the serum was effective in improving the condition of arthritic patient. Use the chi-square test of homogeneity to compare the proportions improved in the populations of treated and untreated subjects. Test at the 5% level of significance.
MA 220
Exercise 6 - Chi-square test
- Is your chance of getting a cold influenced by the number of social contacts you have? A recent study by Sheldon Cohen, a psychology professor at Carnegie Mellon University, seems to show that the more social relationships you have, the less susceptible you are to colds. A group of 266 healthy men and women were grouped according to their number of relationships (such as parent, friend, church member, neighbor). They were then exposed to a virus that causes colds. An adaption of the results is shown in the table: Do the data provide sufficient evidence to indicate that susceptibility to colds is affected
Number of Relationships
Three or Fewer Four or Five Six or More
Cold 49 43 34 No Cold 31 47 62
Total 80 90 96
by the number of relationships you have? Test at the 5% significance level.
