Math 217 Homework 1 Solutions

Math 217 Homework 4 Solutions

9.15

The two-way table of gender by admission is shown below.

	Admit?
Gender	Yes	No	Total
Male	490	310	800
Female	400	300	700
Total	890	610	1500

The admission percentage for males is 100(490/800) = 61.25%, whereas the admission percentage for females is 100(400/700) = 57.14%.

The business school admits 100(400/600) = 66.67% of males and 100(200/300) = 66.67% of females. The law school admits 100(90/200) = 45% of males and 100(200/400) = 50% of females. So within each school, females are admitted at the same or higher rate as males.

When considering only gender and admission rate, it appears that Mountain View favors males slightly (61.25% admission rate, as opposed to 57.14% for females). There is an important confounding variable that is ignored when simply viewing the two-way table of gender and admission. That confounding variable is the school to which the students apply. When you consider the three-way table of gender, admission, and school, you can see that males and females are admitted at the same rate in the business school (66.67%) and females are admitted at a slightly higher rate in the law school (50%, as opposed to 45% of males). That is, females are actually admitted at the same or higher rate when looking at the separate professional schools. Why then does it appear that Mountain View favors males overall? Because the business school is easier to get into—the business school admits 100(600/900) = 66.67% of students, whereas the law school admits 100(290/600) = 48.33%. Furthermore 75% of the males apply to the business school, whereas 57.14% of the females apply to the law school, which is harder to get into. This is an example of what is called Simpson’s Paradox.

9.27

First we want to see if there is a significant relationship between the two variables. The Minitab output from the Chi-square test of independence is shown below. Note that all the expected counts are at least 5, so our P-value should be approximately correct. Assuming there is no relationship between these variables (i.e., they are independent), there is essentially no chance (P-value = 0.000) of getting our data table or a more extreme data table. Hence, we have strong evidence that there is a relationship between these variables.

Chi-Square Test: Education Level and Pet-Ownership Status

Expected counts are printed below observed counts

Chi-Square contributions are printed below expected counts

Pet-Ownership Status

Education Level Non-Pet Owners Dog Owners Cat Owners Total

Less than High School 421 93 28 542

431.46 73.25 37.29

0.253 5.326 2.316

High School Graduate 666 100 40 806

641.61 108.93 55.46

0.927 0.732 4.310

Postsecondary 845 135 99 1079

858.93 145.82 74.25

0.226 0.803 8.254

Total 1932 328 167 2427

Chi-Sq = 23.147, DF = 4, P-Value = 0.000

Now we can investigate the nature of the relationship by looking at row and column percentages. Note that we can use the above table as a guide of where to look, based on the large cell contributions to the chi-square statistic. It appears that there are many more dog owners with less-than-hs education than we would expect under independence. And there are many more cat owners with postsecondary education than we would expect under independence.

The row and column percentages are shown in the two tables below.

	Pet-Ownership Status
Education	Non-Pet	Dog	Cat
< High School	78%	17%	5%
High School	83%	12%	5%
Postsecondary	78%	13%	9%

*Cell Entries are Row Percents

Across all education levels, the highest percentage of people are not pet owners. Separately by education-level, people with less than a high school education own dogs at a slightly higher percentage, and people with postsecondary education own cats at a slightly higher percentage.

	Pet-Ownership Status
Education	Non-Pet	Dog	Cat
< High School	22%	28%	17%
High School	34%	31%	24%
Postsecondary	44%	41%	59%

*Cell Entries are Column Percents

Across all pet-ownership statuses, the highest percentage of people have postsecondary education level. The most prominent feature of this table is that 59% of cat owners have a postsecondary education. In fact, dog owners have less education and cat owners have more education than we would expect if, in fact, there was no relationship between the variables.

9.28

First we want to see if there is a significant relationship between the two variables. The Minitab output from the Chi-square test of independence is shown below. Note that all the expected counts are at least 5, so our P-value should be approximately correct. Assuming there is no relationship between these variables (i.e., they are independent), there is a 24.2% chance of getting our data table or a more extreme data table. Hence, our data are not unlikely under the assumption of no relationship, so we have no significant evidence of a relationship between these two variables. (And when you look at the row percentages—pet-ownership separately by gender—the values are nearly identical.)

Chi-Square Test: Gender and Pet Ownership

Expected counts are printed below observed counts

Chi-Square contributions are printed below expected counts

Pet-Ownership Status

Gender Non-Pet Owners Dog Owners Cat Owners Total

Female 1024 157 85 1266

1008.53 170.60 86.86

0.237 1.085 0.040

Male 915 171 82 1168

930.47 157.40 80.14

0.257 1.176 0.043

Total 1939 328 167 2434

Chi-Sq = 2.838, DF = 2, P-Value = 0.242