Unknown and unequal variance
When we can assume that the two populations are normally distributed and that the
unknown population variances are unequal, an approximate t-test based on independent
random samples is given by:
t =
(X
̅
1
− X
̅
2
)−(μ
1
−μ
2
)
(
s
1
2
n
1
+
s
2
2
n
2
)
1/2
In this formula, we use the tables of the t-
distribution using the ‘modified’ degrees of
freedom. The ‘modified’ degrees of freedom are calc
ulated using the following formula:
df =
(
s
1
2
n
1
+
s
2
2
n
2
)
2
(s
1
2
/n
1
)
2
n
1
+
(s
2
2
/n
2
)
2
n
2

Quantitative Methods
2019 Level I High Yield Notes
© IFT. All rights reserved
42
Hypothesis tests concerning mean differences
(Note: This section has high difficulty and low probability of being tested. Do this section
once you have mastered all remaining topics)
If the samples of the populations whose means we are comparing are dependent, then the
paired comparison test is used. The hypothesis is structured as the difference between
means of two populations.
H
0
: μ
d
= μ
d0
H
a
: μ
d
≠ µ
d0
where:
μ
d
stands for the population mean difference and
μ
d0
stands for the hypothesized value for the population mean difference, which is usually
zero.
In order to arrive at the test statistic, we first determine the sample mean difference using:
d
̅
=
1
n
∑
d
i
n
i=0
The standard error of the mean difference as follows:
s
d
̅
=
s
d
√n
Once we have these two values, we can calculate the test statistic using a t-test. This is
calculated using the following formula using n - 1 degrees of freedom:
t =
d
̅
−μ
d0
s
d
̅
The value of calculated test statistic is compared with the t-distribution values in the usual
manner to arrive at a decision on our hypothesis.
Hypothesis tests concerning variance
(Note: This section has high difficulty and low probability of being tested. Do this section
once you have mastered all remaining topics)
Single population variance
In tests concerning the variance of a single normally distributed population, we use the chi-
square test statistic, denoted by χ
2
.
After drawing a random sample from a normally distributed population, we calculate the
test statistic using the following formula with n - 1 degrees of freedom:
χ
2
=
(n−1)(s
2
)
σ
0
2
where:
n = sample size
s = sample variance
We then determine the critical values using the level of significance and degrees of

Quantitative Methods
2019 Level I High Yield Notes
© IFT. All rights reserved
43
freedom. The chi-square distribution table is used to calculate the critical value.
Two population variance
In order to test the equality or inequality of two variances, we use an F-test which
is the
ratio of sample variances.
The formula for the test statistic of the F-test is:
F =
s
1
2
s
2
2
where:
𝑠
1
2
= the sample variance of the first population with n observations
𝑠
2
2
= the sample variance of the second population with n observations
A convention is to put the larger sample variance in the numerator and the smaller sample
variance in the denominator.