Two Group Means Test

Compare two means from a sample.

DV = continuous,  IV = categorical*

*Can use a continuous IV but must recode it into categories.  Example,  could use income, but need to recode into a few income categories.

The IV is the grouping variable.  There are only 2 groups in a two group means test, so you must choose the two categories of the IV that you want to compare. 

df = n₁ + n₂ - 2

See board for t calc formula.

See board for confidence interval formula.

Example 1.

Medical sociologists argue that children of two parent homes get sick less often than children of one parent homes. To test this theory, you collect a national sample of families, including data on the average number of days children missed school per year and whether the family has one parent or two parent.

Ho: Children of two parent homes get sick as often or more often than children of one parent homes.

H1: Children of two parent homes get sick less often than children of one parent homes.

See board for mathematical hypotheses and diagram.  This formula assumes that the two groups have unequal variances (different standard deviations). There is a different formula if the two groups have equal variances (which is unusual occurrence in the social sciences). 

alpha = .05

1. sub-sample of two parent homes: n = 300, mean = 5, std dev = 2

2. sub-sample of one parent homes: n = 250, mean = 7, std dev = 1.5

df = 300 + 250 - 2 = 548, t crit = 1.64

See board for t calculation.   t calc = -13.33

Reject null. Children of two parent homes get sick less often than children of one parent homes.

Confidence Interval =  (5-7) +/- (1.96*.15) = -1.71 -  -2.29

We are 95% confident that children of two parent homes miss between 1.71 and 2.29 fewer days per year from school than children of one parent homes.

Reject null because null value (0) is not in the interval.

Example 2.

Nationally, fee-for-service (FFS) health insurance plans have different annual costs than preferred provider organizations (PPO).  You collect data from a sample of Wilmingtonians, some of whom have FFS health insurance plans and some of whom have PPOs, to find out how the two plans differ.

Ho: The total health care costs are the same for FFS as for PPOs.

H1: The total health care costs are different for FFS as for PPOs.

alpha = .01

1. sub-sample of FFS patients: n = 1000, mean = 4800, s = 1000

2. sub-sample of PPO patients: n = 1500, mean = 4750, s = 500

df = 1000 + 1500 -2, t crit = 2.58

See board for t calculation and diagram.

t calc = 1.46

Accept null. The total health care costs are the same for FFS as for PPOs.

Confidence Interval =  (4800-4750) +/- (2.58*34.16) =   -38.13 - 138.13

We are 99% confident that the FFS plans costs somewhere between $38.13 less per year to $138.13 more per year than the PPO plan. 

Accept null, because 0 is in the interval.

Example 3.

DV = education, IV = sex, alpha = .05

Null: There is no difference in education between men and women. Mean1 = Mean2.

Research: There is a difference in education between men and women. Mean1 not equal Mean2.

t = 3.19, p=.001

Reject null.  There is a difference in education between men and women.  Men average 13.46 years of education. Women average 13.11 years of education. On average, men get about a semester more of college education than women. 

CI .14 to .57.

0 is not in the interval. Reject null.  

We are 95% confident that in the population of Americans, men get somewhere between .14 to .57 more years of education than women. 

Example 4:  Take Home Exercise

Does race influence socio-economic status?

Race is measured as White or Black.

Socio-economic status is measured as an index of education, income, and occupational prestige. It ranges from 0 to 100.

Whites: Mean SES  = 50.14,  s = 19.38, n = 2118

Blacks: Mean SES = 43.45,  s = 18.43, n = 398

Alpha  = .01

IV = race, 2 categories
DV = SEI, continuous 

Analysis = two group means test

Null Hypothesis:  There is no relationship between race and SES.  White Mean SES = Black Mean SES

Research Hypothesis: There is a relationship between race and SES.  White Mean SES   ≠  Black Mean SES

Two tailed test. 

DF =  2118 +  398 = infinity
t crit = 1.96

t calc = 6.62 (see board for formula)

Reject null.   There is a relationship between race and SES.  On average, the SES among Whites is higher (50.14) than among Blacks (43.45) in the U.S.

Example 5:  Take Home Exercise

Does perceived health status influence how many female sexual partners that people have?

Perceived health status is measured as excellent or good

Number of sexual partners ranges from 0 to 989.

Alpha = .05

Excellent Health:  Mean # of female sexual partners = 9.05, s = 47.92, n = 529

Good Health:   Mean # of female sexual partners = 7.30, s = 24.76, n = 812

t calc = .78, p = .44

IV = perceived health status, excellent or good (2 categories)
DV = # of female sexual partners, continuous

Analysis = two group means test

Null Hypothesis: Perceived health status does not influence how many female sexual partners that people have.  Mean # of Female Sexual Partners Among People with Excellent Health  =  Mean # of Female Sexual Partners Among People with Good Health

Research Hypothesis: Perceived health status does not influence how many female sexual partners that people have.  Mean # of Female Sexual Partners Among People with Excellent Health  ≠  Mean # of Female Sexual Partners Among People with Good Health

Two tailed test

p is higher than alpha.  Accept null.  We would be wrong approximately 44% of the time (if we took repeated samples) if we rejected the null.

Interpretation:  Perceived health status does not influence how many female sexual partners that people have.  People who perceive themselves to be in excellent health have about the same number of female sexual partners in their lifetimes as people who perceive themselves to be in good health.  On average the number of female sexual partners among people in good or excellent perceived health is somewhere between 7 and 9 women.