Crosstabs, Means Tests, and Correlation Exercises: alpha = .05 on all tests

 

 

1. Is there a relationship between the number of police officers on staff and the type of crimes that are most frequent?

 

 

 

1998, 5 officers

1999, 8 officers

Property Crime Arrests

325

375

Violent Crime Arrests

120

119

 

Analysis = Crosstabs w/ chi-square

 

Ho: There is no relationship between the number of officers and the number of property or violent crime arrests.   (chi square = 0).

H1: There is a relationship between the number of officers and the number of property or violent crime arrests.   (chi square not equal to 0). 

 

Df = 1*1 = 1

X crit = 3.84

 

Expected Values

 

 

1998, 5 officers

1999, 8 officers

Totals

Property Crime Arrests

331.74

368.26

700

Violent Crime Arrests

113.26

125.74

235

Totals

445

494

939

 

 

Calculation of chi-square

 

f(o)

f(e)

fo-fe

(fo-fe) (fo-fe)

[(fo-fe) (fo-fe)]/fe

325

331.74

-6.74

45.43

.14

120

113.26

6.74

45.43

.40

375

368.26

6.74

45.43

.12

119

125.74

-6.74

45.43

.36

 

 

 

Sum

1.02

 

 

Chi-square = 1.02

 

Accept null.

 

To interpret what this means, you need to consult the observed table again, and look at the percentage responses (which you may need to calculate).

 

 

1998, 5 officers

1999, 8 officers

Totals

Property Crime Arrests

325 (73%)

375 (76%)

700

Violent Crime Arrests

120 (27%)

119 (24%)

235

Totals

445

494

939

 

 

Interpretation:  There is no relationship between the # of officers and the number of property or violent crime arrests.  Similar percentages of property crime arrests and violent crime arrests occurred in 1998 when there were 5 officers as in 1999 when there were 8 officers.

 

 

 

2.      Is there a relationship between race and attitude about whether whites were hurt by affirmative action?  Be sure to use the statistics that are provided in the information below.

 

Number of Whites = 2238

Number of Blacks = 432

% White who think it is very likely or somewhat likely that whites were hurt by affirmative action: 75.8%
% Black who think it is very likely or somewhat likely that whites were hurt by affirmative action: 50.4%

p = .000, chi-square = 171.47

Analysis = Crosstabs w/ chi-square

Ho: There is no relationship between race and and attitudes about whether whites were hurt by affirmation action (chi square = 0).

H1: There is a relationship between race and attitudes about whether whites were hurt by affirmation action (chi square ≠ 0).

Compare p to alpha.  p is less than alpha. Reject null.

To interpret what this means, you need to consult the observed table again, and look at the percentage responses (which you may need to calculate).

Intepretation:  There is a relationship between race and attitude about whether whites were hurt by affirmative action (chi-square = 37.31, p<.000).  More whites (75.8%) than blacks (50.4%) think it is very likely or somewhat likely that whites were hurt by affirmative action.

 

3. I think being a feminist or not influences whether people think homosexuals should be able to teach in public schools.  Use the data below, from the GSS96, to test my hypothesis.  

 

Identify as a Feminist

 

Allow Homosexuals to Teach?

Yes

No

Totals

Yes

156

508

664

No

33 

172

205

Totals

189

680

869

Chi-square = 5.04, p = .025

Analysis = Crosstabs w/ chi-square

Ho:  Being a feminist does not influence beliefs on whether homosexuals should be able to teach school. (chi square = 0).

H1: Being a feminist influences beliefs on whether homosexuals should be able to teach school. (chi square ≠ 0).

Compare p to alpha.  P is less than alpha.  Reject null.  

To interpret what this means, you need to consult the observed table again, and look at the percentage responses (which you may need to calculate).

 

 

 

Identify as a Feminist

 

Allow Homosexuals to Teach?

Yes

No

Totals

Yes

156 (83%)

508 (75%)

664

No

33 (17%)

172 (25%)

205

Totals

189

680

869

 

Interpretation: Being a feminist influences beliefs on whether homosexuals should be able to teach school. Feminists are more likely to think that homosexuals should be allowed to teach school (83%) than people who are not feminist (75%).

 

4. I think being a feminist or not influences whether people think racists should be able to teach in public schools.  Use the data below, from the GSS96, to test my hypothesis.

 

 

Identify as a Feminist

 

Allow Racists to Teach?

Yes

No

Totals

Yes

104

306

410

No

82

363

445

Totals

186

669

855

Chi-square = 6.04, p = .014

Analysis = Crosstab w/ chi square

Ho:  Being a feminist does not influence beliefs on whether racists should be able to teach school. (chi square = 0).

H1: Being a feminist influences beliefs on whether racists should be able to teach school. (chi square ≠ 0).

compare p to alpha.  P is less than alpha.  Reject null.  

To interpret what this means, you need to consult the observed table again, and look at the percentage responses (which you may need to calculate).

 

 

 

Identify as a Feminist

 

Allow Racists to Teach?

Yes

No

Totals

Yes

104 (56%)

306 (46%)

410

No

82 (44%)

363 (54%)

445

Totals

186

669

855

 

Intepretation:  Being a feminist influences beliefs on whether racists should be able to teach school. Feminists are more likely to think that racists should be allowed to teach school (56%) than people who are not feminist (46%).

 

5. I think that people with a college degree tend to have fewer children than people without a college degree.  Use the data below, from the GSS96, to test my hypothesis.  

People with a college degree: n = 694, mean # of children = 1.41, s = 1.5

People without a college degree: n = 2188, mean # of children = 1.97, s = 1.71

Analysis = Two Group Means Test

Ho: The average number of children among people with a college degree is greater than or equal to the average number of children among people without a college degree. (mean 1 ≥ mean 2).

 

H1: The average number of children among people with a college degree is less than the average number of children among people without a college degree (mean 1 < mean 2).

 

df = 694 + 2188 –2  = use df at infinity:   t crit =  1.64

 

Draw diagram.  See board.

 

T calc = -9.33

 

Numerator = 1.41 – 1.97 = -.56

 

Denominator = sq rt of [(1.52/694) + (1.712/2188)]  = sq rt of .003 + .001

= -.56/.06 = -9.33

 

t calc is greater than t crit.  Reject null.

 

Intepretation: People with a college degree tend to have fewer children (avg = 1.41) )than people without a college degree (avg = 1.97).

 

6. I think people who hold progressive beliefs and values concerning gender differ in their amount of happiness (on a scale that ranges from 0-7) from other Americans.  Use the data below, from the GSS96, to test my hypothesis.  Be sure to use the statistics that are provided in the information below.

 

 

People who hold progressive gender beliefs and values: n = 672, mean happiness = 5.13, s = 2.11

Mean happiness score of all Americans:  5.27

t calc = -1.69, p = .09

Analysis = One Group Means Test

Ho: People with progressive beliefs about gender have the same level of happiness as all Americans.  (mean = 5.27) 

H1: People with progressive beliefs about gender have a different level of happiness as all Americans.  (mean ≠ 5.27) 

 

Compare p to alpha.  No calculations necessary.  p is bigger than alpha.  Accept null.

 

Interpretation: People with progressive beliefs about gender have the same level of happiness as all Americans.

 

7.      I think that romantic partners often have different levels of satisfaction with their relationship. I collect a sample of 9 romantic partners and ask each partner how satisfied they are with their relationship (on a scale of 0-10).  Use the data below to test my hypothesis.  

 

Partner 1 Satisfaction

Partner 2 Satisfaction

Difference

8

5

3

7

2

5

8

8

0

7

7

0

10

9

1

5

4

1

2

4

-2

7

7

0

5

6

-1

 

Mean difference = .78,  standard deviation of the difference = 1.99

Analysis = Matched Group Means Test

Ho:  Romantic partners have the same level of satisfaction with their relationships (d = 0).

H1: Romantic partners do not have the same level of satisfaction with their relationships (d ≠ 0).

Two tailed test.  t crit = 2.31 (+/-) (df = 9-1)

t calc =  1.18  

t = .78 / (1.99/sq rt of 9)

t = .78/(1.99/3)

t=.78/.66

t =1.18

 

t calc is less than t crit.  Accept null. 

Interpretation: Romantic partners have the same level of satisfaction with their relationships.

 

7.      I think that education influences income. Use the data below to test my hypothesis.

 

Group

Sample Size

Mean Income

Std. Dev.

Less than H.S. degree

30

15,545

165.21

H.S. Degree

120

19,621

342.87

Some College

40

20,210

155.99

Bachelor’s Degree

80

24,398

899.34

Graduate Degree

15

33,568

8003.84

F= 14.27, p=.000

Analysis = multiple group means test (ANOVA)

Ho: There is no relationship between education and income.  Mean1= mean 2 = mean3 = mean4 = mean5, f=0

(or you could say, education does not influence income)
H1: There is a relationship between education and income. Mean1 ≠ mean 2 ≠ mean3 ≠ mean4 ≠ mean5, f ≠ 0

(or you could say, education does influence income)

p is less than alpha.  Reject the null.

Interpretation: Education does influence income. As education increases, people tend to earn more income.  People with a graduate degree, on average, have the highest income ($33,568), and people with less than a high school degree, on average, have the lowest income ($15,545).

 

8.      Is there a relationship between SEI and number of hours worked last week?

r = .23, p = .09 (two tailed)

analysis = correlation

Ho: There is no relationship between SEI and the number of hours worked. r=0
H1: There is a relationship between SEI and the number of hours worked. r≠0

p is higher than alpha.  Accept the null.

Interpretation: There is no linear relationship between SEI and the number of hours that people work.

 

What if this had been a one tailed test?  What if you hypothesized a positive correlation?

Ho: There is either no relationship between SEI and the number of hours worked or a negative relationship ( r=0 or r<0)
H1: As the number of hours worked increases, SEI increases ( r > 0).

All error goes on the right hand side of  your sampling distribution.

 

Divide p from above in half (p = .045)

 

P is less than alpha.  Reject null. 

r*r = .23 *.23 = .05  

 

Interpretation: There is a moderate, positive correlation between hours worked and SEI.  As the number of hours worked increases, SEI increases.  The number of hours worked explains 5% of the variation in SEI.

 


 

10. Is there a relationship between years of education and dollars earned?

r = .33, p = .000 (two tailed)
analysis = correlation

 

Ho: There is no relationship between years of education and dollars earned. r=0
H1: There is a relationship between years of education and dollars earned. r≠0

p is lower than alpha.  Reject the null.

r2 = .1089

Interpretation: There is a moderate positive relationship between SEI and the number of hours that people work.  As education increases, people’s income tends to increase.  Education explains 10.89% of the variation in income.