17 - 1 Module 17: Two-Sample t-tests, with equal variances for the two populations This module describes one of the most utilized statistical tests, the two-sample t-test conducted under the assumption that the two populations from which the two samples were selected have the same variance. Reviewed 11 May 05 /MODULE 17
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17 - 1
Module 17: Two-Sample t-tests, with equal variances for the two
populations
This module describes one of the most utilized statistical tests, the two-sample t-test conducted under the assumption that the two populations from which the two samples were selected have the same variance.
Reviewed 11 May 05 /MODULE 17
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Up to this point, the focus has been on a single population, for which the observations had a normal distribution with a population mean μ and standard deviation σ. From this population, a random sample of size n provided the sample statistics and s as estimates of μ and σ, respectively.
We created confidence intervals and tested hypotheses concerning the population mean μ, using the normal distribution when we had available the value of σ and using the t distribution when we did not and thus used the estimate s from the sample. This circumstance is often described as the one sample situation.
x
The General Situation
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Clearly, we are often faced with making judgments for circumstances that involve more than one population and sample. For the moment, we will focus on the so-called two sample situation. That is, we consider two populations.
Question:
Do you believe the two populations have the same mean?
σBσASD
µBµAMean
City BCity A
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H0: μA = μB versus H1: μA ≠ μB
or equivalently
H0: Δ = μA - μB = 0 versus H1: Δ = μA - μB ≠ 0.
Two Sample Hypotheses
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Population 1 Population 2 Parameter Estimate Parameter Estimate
Populations of individual values
μ1 μ2
σ12 s1
2 σ22
s22
σ1 s1
σ2 s2
Populations of means, samples of size n1 and n2
μ1 μ2
σ12/n1 s1
2/n1 σ22/n2
s22/n2
σ1/√n1 s1/√n1 σ2/√n2 s2/√n2
1x 2x
2x1x
Parameters vs. Estimates
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We are interested in
Δ = µ1 - µ2
If the samples are independent, then
When
1 2d x x= −
1 2 1 22 2
1 21 2
1 2
( ) ( ) ( )
( )
V a r x x V a r x V a r x
V a r x xn nσ σ
− = +
− = +
2 2 21 2 1 2
1 2
1 1, ( )Var x xn n
σ σ σ⎛ ⎞
= − = +⎜ ⎟⎝ ⎠
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we have two estimates of σ2 , one from sample 1, namely s1
2 and one from sample 2, namely s22. How
can we best use these two estimates of the same thing. One obvious answer is to use the average of the two; however, it may be desirable to somehow take into account that the two samples may not the same size. If they are not the same size, then we may want the larger one to count more.
2 2 21 2When ,σ σ σ= =
Estimating σ2
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Hence, we use the weighted average of the two sample variances, with the weighting done according to sample size. This weighted average is called the pooled estimate:
( ) ( )( ) ( )11
11
21
222
2112
−+−−+−
=nn
snsnsp
Pooled Average
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To estimate Var( ), we can use
⎟⎟⎠
⎞⎜⎜⎝
⎛+
21
2 11nn
sp
1 2x x−
1 2x x−Estimate of Var( )
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Statistic City A City B N 10 10 x (mmHg) 105.8 97.2 s2(mmHg)2 78.62 22.40 s (mmHg) 8.87 4.73
To investigate the question of whether the children of city A and city B have the same systolic blood pressure, a random sample of n = 10 children was selected from each city and their blood pressures measured. These samples provided the following data:
Example 1: Blood Pressures of Children
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We are interested in the difference:
Δ = μA - μB
and we have as an estimate of μA and as an estimate of μB; hence it is reasonable to use:
d = - = 105.8 - 97.2 = 8.6 (mm Hg)
as an estimate of Δ = μA - μB.
Ax Bx
Ax Bx
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We then can ask whether this observed difference of 8.6 mm Hg is sufficiently large for us to question whether the two population means could be the same, that is, μA = μB. Clearly, if the two population means are truly equal, that is, if μA = μB is true, then we would expect the two sample means also to be equal, that is = , except for the random error that occurs as a consequence of using random samples to represent the entire populations. The question before us is whether this observed difference of 8.6 mm Hg is larger than could be reasonably attributed to this random error and thus reflects true differences between the population means.