One Group Mean Test
Compare sample mean to a population mean.
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 in a one group means test is the grouping variable. Since there is only one group, there really isn't an IV in a one group means test. But you are comparing a sample group to a population group. The sample vs. the population is the IV.
t calc = (sample mean - population mean) / (standard
deviation/square root of n)
see board
t crit = look up at df = n-1
Hypotheses: write both in terms of population mean; can have directional hypotheses
Confidence Interval = sample mean +/- two tailed t crit (s /sq rt of n)
See board
Example 1.
Past research shows that Asian Americans do better on the math SAT than other Americans. The national average on the math SAT is 500. We have the math SAT scores from a sample of Asian American students and want to know if they are still doing better than other students.
Ho: Asian Americans do as well or less well on the math SAT as other Americans.
H1: Asian Americans do better on the math SAT than other Americans.
Alpha = .05
See board for mathematical hypotheses and diagram.
n = 400, sample mean = 507, std dev = 100
df = 399, t-crit = 1.64
t calc = (507-500)/(100/sq rt of 400) = 1.4
see board for t calculation
Decision: Accept null. Asian Americans do the same on the math SAT as other Americans.
Confidence Interval = 507 +/- 1.96 (100/sq rt of 400) = 497.2 - 516.8
See board for confidence interval calculation.
We are 95% confident that in the population of Asian Americans taking the math SAT, the average score is somewhere between 497.2 and 516.8.
Accept null because null value is in interval.
Example 2. On the Computer
DV = education, test value (population
parameter) = 12, alpha =.05
Null: People today on average are getting a high school degree or
less. mean lt or equal 12
Research Hypothesis: People today on average are getting more education than a high school degree. Mean gt 12
t = 23.36, p = .000
Reject Null. People today on average are getting more education than a high school degree. Today, people average 13.26 years of education -- about 1 and a quarter year of college.
CI = 1.16 - 1.37
0 is not in the interval -- reject null (0 would mean no difference between the sample mean and population parameter. In this case it means the sample mean was equal to 12. 12-12 = 0)
We are 95% confident that in the population of all Americans people have somewhere between 1.16 to 1.37 years of education beyond high school.
Take Home Exercise 1
I think the number of siblings that Americans have has decreased since 1940. In 1940 the average number of siblings among adult Americans was 4.2.* From the 2000 GSS, we know the following about the number of siblings today:
Mean = 3.67, s = 2.99, n = 2806
* This is fictitious.
IV = year, 2000 vs 1940
DV = # of siblings
Analysis = one group means test, because there is only data on one group
Null: The number of siblings that Americans have has stayed the same or increased since 1940.
Research: The number of siblings that Americans have has decreased since 1940.
t crit = -1.64
t calc = -9.42 (or something close to that)
Reject null. The number of siblings that Americans have has decreased since 1940. People today have an average of 3.67 siblings, which is statistically lower than 4.2 siblings in 1940.
Take Home Exercise 2
I think Southerners differ from other Americans in how many hours of TV they watch per day. The average number of TV hours viewed daily among all Americans is 2.97 hours. Using the GSS, I looked at just respondents who live in the South and found the following:
Mean TV Hours Among Southerners = 3.09, s = 2.59, n = 651, t = 1.16, p = .25
IV = Southerners vs. all Americans
DV = # of TV hours
Analysis = one group means test (only data on one group)
Null: Southerners do not differ from other Americans in how many hours of TV they watch per day.
Research: Southerners differ from other Americans in how many hours of TV they watch per day.
t calc = 1.16, p = .25
Accept null.
Southerners do not differ from other Americans in how many hours of TV they watch per day. Southerners watch on average 3.08 hours of TV a day, which is not statistically different than the national average of 2.97 hours a day.