Download The sampling distribution for ¯x1 − ¯x2 We assume that we have a

Chapter 24
The sampling distribution for x̄1 − x̄2
We assume that we have a SRS of size n1 of independently
selected measurements of the quantity x1 from a population with population mean value µ1 and standard deviation
σ1, and another SRS of size n2 of independently selected
measurements of the quantity x2 from a population with
population mean value µ2 and standard deviation σ2. Both
samples satisfy the 10% Condition as well.
• the mean of the sampling distribution of differences
x̄1 − x̄2 is µ1 − µ2;
• the variance of the sampling distribution of differences
is the sum of the variances of the individual proportion variables; therefore,the standard deviation of the
sampling distribution of x̄1 − x̄2 is
s
σ12 σ22
+ .
SD(x̄1 − x̄2) =
n1 n2
• since the population parameters σ1 and σ2 are typically
unknown, we estimate the standard deviation with the
standard error
s
SE(x̄1 − x̄2) =
1
s21 s22
+ .
n1 n2
Chapter 24
The two-sample t-interval
A level C confidence interval for x̄1 − x̄2 is
(x̄1 − x̄2) − t∗df ·SE(x̄1 − x̄2)
≤ µ 1 − µ2 ≤
(x̄1 − x̄2) − t∗df · SE(x̄1 − x̄2),
or, more compactly,
µ1 − µ2 = (x̄1 − x̄2) ± t∗df · SE(x̄1 − x̄2),
where t∗df is the appropriate critical value of t for the
given level of confidence, and its degrees of freedom df
is given by the formula
2
2
s22
s1
n1 + n2
df =
2 2
2 2 (!!)
s1
s2
1
1
+
n1 −1 n1
n2 −1 n2
Of course, it is best to let the entire confidence interval
be computed by your calculator (or other software).
[TI-83: STAT TESTS 2-SampTInt]
2
Chapter 24
The Two-sample t hypothesis test
• State hypotheses:
Null hypothesis: H0 : µ1 = µ2
Alternative hypothesis: HA : µ1 > µ2; or µ1 < µ2; or
µ1 6= µ2
• Choose model:
Two independent SRS are selected from nearly normal
populations satisfying the 10% Condition, one of size
n1 and another of size n2, so Student’s t model with
df = (n1 − 1) + (n2 − 1) degrees of freedom applies for
the standardized sampling distribution for the statistic
x¯1 − x¯2
.
tdf = q 2
s22
s1
n1 + n2
• Mechanics:
Compute tdf statistic; then find P -value associated with
the respective form of HA: P = P (T ≥ tdf | H0); or
P = P (T ≤ tdf | H0); or P = 2P (T ≥ tdf | H0).
• Conclusion:
Assess evidence to reject (or fail to reject) H0 depending
on how small P is.
[TI-83: STAT TESTS 2-SampTTest]
3
Chapter 24
Tukey’s test for comparing means
• State hypotheses:
Null hypothesis: H0 : µ1 = µ2
Alternative hypothesis: HA : µ1 > µ2 (Note: only one
alternative option)
• Choose model:
Two independent SRS are selected from nearly normal
populations satisfying the 10% Condition, one of size
n1 and another of size n2; the first sample contains the
largest of all the measurements and the second contains
the smallest.
• Mechanics:
Count how many measurements in the first sample are
greater than all measurements in the second, count how
many measurements in the second sample are less than
all measurements in the first, then add the two counts to
obtain the Tukey statistic τ (‘tau’, a Greek t), counting
any ties as worth 1/2.
• Conclusion:
Reject H0 in favor of HA at α = 0.05 if τ ≥ 7;
reject H0 in favor of HA at α = 0.01 if τ ≥ 10;
reject H0 in favor of HA at α = 0.001 if τ ≥ 13.
4