Other assumptions required for this procedure to be

Other assumptions required for this procedure to be valid are:
•
•
For both the confidence interval and hypothesis testing procedures, we are interested in the difference
between population means: μ
1
− μ
2
.
If X
¯
1
− X
¯
2
is a statistic, then it must have a sampling distribution. If the assumptions above are met, then
the sampling distribution for X
¯
1
− X
¯
2
is:
10.1.1 Confidence Intervals for the Difference in Population Means
Using the information about the sampling distribution from above, a (1 − α )100% confidence interval
for the difference in population means will be of the form:
Our interpretation of this confidence interval is the same as what we have done for the one population
examples:
10.1.2 Hypothesis Tests for the Difference in Population Means
Our null hypothesis takes the general form H
0
: μ
1
− μ
2
= D
0
, where D
0
is some hypothesized difference
between the population means.Often, we are testing the null hypothesis of no difference , or no effect .

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There are three options for the alternative hypothesis:
• H
a
: μ
1
6= μ
2
(two-sided test)
• H
a
: μ
1
< μ
2
(one-sided, lower/left-tail test)
• H
a
: μ
1
> μ
2
(one-sided, upper/right-tail test) The test statistic is calculated as:
If H
0
is true and the assumptions are satisfied, Z will follow a standard normal distribution.
Graphically:
EXAMPLE: A study was conducted to compare the starting salaries of university graduates from an
honours Bachelor’s program, versus those graduates in a general Bachelor’s program. Previous studies
had found that the population standard deviation of starting salaries for honours and general graduates
were $2250 and $2380 respectively, while independent random samples of size 50 from each
population resulted in sample mean starting salaries of $45,554 and $43,348 respectively. It is
reasonable to assume that the starting salaries of both honours and general degree graduates are
normally distributed.
a) Does this data provide sufficient evidence, at the 5% level of significance, to suggest that
honoursgraduates make higher mean starting salaries than general graduates?

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b) What is a 95% confidence interval for the true difference in mean starting salaries between the
twogroups of graduates?
10.2 Two-sample procedures when σ
1
and σ
2
are UNKNOWN
Usually we do not know σ
1
and σ
2
, in which case we have two options for inference procedures on two
population means:
• Welch’s approximate t procedure: we don’t know σ
1
and σ
2
, and do not assume they are equal.
• Pooled variance t procedure: even though we don’t know σ
1
and σ
2
, we assume they are equal.
10.2.1 Welch’s Approximate t procedure
This method requires two assumptions in order to be valid:
•
•
In Welch’s procedure, we simply substitute the sample variances, and , in for the values of and
:
The (1 − α )100% confidence interval for μ
1
− μ
2
is then given by:
In the hypothesis testing procedure, our test statistic becomes:

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If the null hypothesis is true, then the test statistic follows an approximate t distribution.