sociology 5811: t-tests for difference in means wes longhofer, pinch-hitting for evan schofer
TRANSCRIPT
Sociology 5811:T-Tests for Difference in Means
Wes Longhofer, pinch-hitting for Evan Schofer
Strategy for Mean Difference
• We never know true population means• So, we never know true value of difference in means
• So, we don’t know if groups really differ
• If we can figure out the sampling distribution of the difference in means…
• We can guess the range in which it typically falls
• If it is improbable for the sampling distribution to overlap with zero, then the population means probably differ
• An extension of the Central Limit Theorem provides information necessary to do calculations!
Sampling Distribution for Difference in Means
• The mean (Y-bar) is a variable that changes depending on the particular sample we took
• Similarly, the differences in means for two groups varies, depending on which two samples we chose
• The distribution of all possible estimates of the difference in means is a sampling distribution!
• The “sampling distribution of differences in means”
• It reflects the full range of possible estimates of the difference in means.
Mean Differences for Small Samples
• Sample Size: rule of thumb
• Total N (of both groups) > 100 can safely be treated as “large” in most cases
• Total N (of both groups) < 100 is possibly problematic
• Total N (of both groups) < 60 is considered “small” in most cases
• If N is small, the sampling distribution of mean difference cannot be assumed to be normal
• Again, we turn to the T-distribution.
Mean Differences for Small Samples
• To use T-tests for small samples, the following criteria must be met:
• 1. Both samples are randomly drawn from normally distributed populations
• 2. Both samples have roughly the same variance (and thus same standard deviation)
• To the extent that these assumptions are violated, the T-test will become less accurate
• Check histogram to verify!
• But, in practice, T-tests are fairly robust.
Mean Differences for Small Samples
• For small samples, the estimator of the Standard Error is derived from the variance of both groups (i.e. it is “pooled”)
• Formulas:
2
))(1())(1(s
21
222
211
)Y-Y( 21
NN
sNsN
Probabilities for Mean Difference
• A T-value may be calculated:
21
212
1121
21
NNs
)YY(t
)YY(
)N(N
• Where (N1 + N2 – 2) refers to the number of degrees of freedom– Recall, t is a “family” of distributions– Look up t-dist for “N1 + N2 -2” degrees of freedom.
T-test for Mean Difference
• Back to the example: 20 boys & 20 girls
• Boys: Y-bar = 72.75, s = 8.80
• Girls: Y-bar = 78.20, s = 9.55
• Let’s do a hypothesis test to see if the means differ:
• Use -level of .05
• H0: Means are the same (boys = girls)
• H1: Means differ (boys ≠ girls).
T-test for Mean Difference
21
212
1121
21
NNs
)YY(t
)YY(
)N(N
201
201
45.5
21
38
)YY(
)(
s
)(t
• Calculate t-value:
T-Test for Mean Difference
• We need to calculate the Standard Error of the difference in means:
2
))(1())(1(s
21
222
211
)Y-Y( 21
NN
sNsN
38
)55.9)(19()80.8)(19(s
22
)Y-Y( 21
T-Test for Mean Difference
• We also need to calculate the Standard Error of the difference in means:
38
)85.1732()36.1471(s )Y-Y( 21
18.932.84s )Y-Y( 21
T-test for Mean Difference
21
212
1121
21
NNs
)YY(t
)YY(
)N(N
201
201
)18.9(
45.538
)(t )(
• Plugging in Values:
T-test for Mean Difference
)316)(.18.9(
45.538
)(t )(
88.1)90.2(
45.538
)(t )(
T-Test for Mean Difference
• Question: What is the critical value for =.05, two-tailed T-test, 38 degrees of freedom (df)?
• Answer: Critical Value = approx. 2.03
• Observed T-value = 1.88
• Can we reject the null hypothesis (H0)?
• Answer: No! Not quite!• We reject when t > critical value
T-Test for Mean Difference
• The two-tailed test hypotheses were:
GirlsBoys μμ :H1
GirlsBoys μμ :H0
• Question: What hypotheses would we use for the one-tailed test?
GirlsBoys μμ :H1
GirlsBoys μμ :H0
T-Test for Mean Difference
• Question: What is the critical value for =.05, one-tailed T-test, 38 degrees of freedom (df)?
• Answer: Around 1.684 (40 df)
• One-tailed test: T =1.88 > 1.684• We can reject the null hypothesis!!!
• Moral of the story:• If you have strong directional suspicions ahead of time, use
a one-tailed test. It increases your chances of rejecting H0.
• But, it wouldn’t have made a difference at =.01
Another Example• Question: Do the mean batting averages for American
League and National League teams differ?• Use a random sample of teams over time
• American League: Y-bar = .2677, s = .0068, N=14• National League: Y-bar = .2615, s = .0063, N=16• Let’s do a hypothesis test to see if the means differ:
• Use -level of .05
• H0: Means are the same (American = National)
• H1: Means differ (American ≠ National)
T-test for Mean Difference
21
212
1121
21
NNs
)YY(t
)YY(
)N(N
161
141
0062.
21
28
)YY(
)(
s
)(t
• Calculate t-value:
T-Test for Mean Difference
• We need to calculate the Standard Error of the difference in means:
2
))(1())(1(s
21
222
211
)Y-Y( 21
NN
sNsN
28
)0063)(.15()0068)(.13(s
22
)Y-Y( 21
T-Test for Mean Difference
• We also need to calculate the Standard Error of the difference in means:
28
)0006(.)0006(.s )Y-Y( 21
0065.s )Y-Y( 21
T-test for Mean Difference
21
212
1121
21
NNs
)YY(t
)YY(
)N(N
161
141
)0065(.
0062.)28(
)(t
• Plugging in Values:
T-test for Mean Difference
)366)(.0065(.
0062.28
)(t )(
58.2)0024(.
0062.28
)(t )(
T-Test for Mean Difference
• Question: What is the critical value for =.05, two-tailed T-test, 28 degrees of freedom (df)?
• Answer: Critical Value = approx. 2.05
• Observed T-value = 2.58
• Can we reject the null hypothesis (H0)?
• Answer: Yes• We reject when t > critical value
• What if we used an -level of .01?– Critical value=2.76
T-Test for Mean Difference• Question: What if you wanted to compare 3 or
more groups, instead of just two?• Example: Test scores for students in different educational
tracks: honors, regular, remedial
• Can you use T-tests for 3+ groups?• Answer: Sort of… You can do a T-test for every
combination of groups• e.g., honors & reg, honors & remedial, reg & remedial
• But, the possibility of a Type I error proliferates… 5% for each test
• With 5 groups, chance of error reaches 50%• Solution: ANOVA.