Workshop Statistics: Discovery with Data and Fathom

Topic 26: Inference for Two-Way Tables

Activity 26-1: Suitability for Politics (cont.)

(a) observational units = sample of American adults
variable 1: whether or not they agree with the statement; categorical binary
variable 2: their political view: categorical
(b) Tend to have more agreement with statement as lean more conservative. While there is definitely a trend it may not be all that strong.
(c) agreement proportion = 382/1685 = .227
(d) Liberals: 484(.227) =109.87
(e) Moderates: 610(.227)=138.47
Conservatives: 591(.227)=134.16
(f) 382(484)/1685=109.73
(g) rest of row is 138.47 and 134.16
(h) (74-109.87)2/109.87 = 11.632
(139-138.47)2/138.47 = .004
(169-134.16)2/134.16 =9.152
(410-374.27)2/374.27 =3.140
(471-471.71)2/471.71 = .001
(422-457.02)2/457.02 = 2.683
sum = 26.881
(i) large values of the test statisitc will be evidence against the null hypothesis (indicate larger discrepancies between the observed and the expected)
(j) df = (3-1)(2-1) = 2
26.881 is off the chart so p-value is < .0005
(k) It would be highly unusual to observe these sample data by chance alone if there was independence between agreement and political leaning. We have very strong evidence that political leaning is related to people’s agree with the statement "Most men are better suited emotionally for politics than are
most women."

Activity 26-2: Government Spending

(a) Ho: There is no association between opinionon spending and political leaning
Ha: There is an association between opinionon spending and political leaning
(b) 133(376)/1235 = 40.49
590(435)/1235 = 207.81
512(424)/1235 = 175.78
(c) (40.49-50)2/50 = 2.232
(214-207.81)2/207.81 = .184
(176-175.78)2/175.78 = .000
 sum=6.724
(d) with df=(3-1)(3-1)=4, find .1 < p-value < .2
(e) There is no evidence of an association between opinion and spending on space program.

Activity 26-3: Newspaper Reading (cont.)

(a)

Ho: There is no association between gender and newspaper reading
Ha: There is an association between gender and newspaper reading
test statistic: 8.261, p-value = .041
We would reject the null hypothesis at the .05 level and conclude there is a relationship between gender and newspaper reading. (We would not reject at the .01 level.)

(b)

    The shaded region represents the probability of seeing a test statistic as large as the one observed (8.261) when there is no association.  Since this probability is small (less than .05) we make conclude that there probably is some association between gender and newspaper reading..

(c)  The cell that corresponds to the male, reading less than once per week contributes the most to the test statistic. The observed count is lower than the expected count for that cell, meaning that there are fewer males reading less than once per week than we would’ve expected if there was no relationship.
(d) The next three cells are the cells that correspond to men reading every day, women reading every day, and women reading less than once per week.  There are more men than we would have expected reading every day, there are fewer women then we would expected reading every day and more women
than expected reading less than once per week. Seems men read more than women.

Activity 26-4: Suitability for Politics (cont.)

(a)

Ho: There is no association between gender and suitability opinion
Ha: There is an association between gender and suitability opinion
test statistic: .01776, p-value: .89
With our large p-value, there is no evidence of an association between gender and their opinion on whether men are more suitable for politics than women.

(b) Let q1=proportion of men who agree with the statement and q2=proportion of women who agree with the statement
Ho: q1=q2 (men and women agree in equal proportions)
Ha: q1¹q2(the proportions of men and women who agree differs)
test statistic: -.13, p-value: .894
Again, with this large p-value, we would fail to reject Ho: q1=q2, giving us no evidence that men and women don't agree in the same proportion.
(c) The p-values are equal, and the conclusion is the same.
(d) z=-.13, z2=.0169 which is close to .018.
 

Activity 26-5: Government Spending (cont.)

Answers will vary. These are sample answers.
(b) Sample output:
liberal moderate conservative total
too little 45 43 45 133
just right 169 224 197 590
too much 162 168 182 512
total 376 435 424 1235

Chi-square statistic for this table: 4.174

(c) This is smaller than 6.724, the chi-square statistic from the actual sample data.
(d)

(e) There were 11 values in this graph larger than 6.725. This is 11% of the repetitions.
(f) Earlier we found p-value to be between .1 and .2, and .11 is within this range, so yes, we are reasonable close.  They should match since the p-value tells us how often we expect to get a chi-square value at least this extreme when the null hypothesis is true.