2. So far, the topics we have covered are:
• Test of mean of a normal distribution: population variance known
• One tailed and two tailed tests
• P value
• Type I and Type II error
• Power of test
• Test of mean of a normal population: Population variance unknown
• Test of population proportion
• Hypothesis test of variance of a normal population
Today we shall look at
• Testing difference between means of two independent population
• Testing equality of variances of two normal population
• F distribution
3. Tests for difference between two means (Independent samples)
• There are a number of applications where we wish to draw conclusions about the differences
between population means instead of conclusions about the absolute levels of the means.
• For example, we might want to compare the output of two different production processes for
which neither population mean is known. Similarly, we might want to know if one marketing
strategy results in higher sales than another without knowing the population mean sales
for either.
• These questions can be handled effectively by various different hypothesis-testing
procedures.
• Suppose there is a random sample of 𝑛𝑥from a normal population with mean 𝜇𝑥 and variance 𝜎𝑥
2
and an independent sample of observations 𝑛𝑦 from a normal population with mean 𝜇𝑦and
variance 𝜎𝑦
2. The observed sample means are 𝑥 and 𝑦 and significance level is α.
4.
5.
6. Example 1
A political science professor is interested in comparing
the characteristics of students who do and do not vote in national elections.
For a random sample of 114 students who claimed to have voted in the last
presidential election, she found a mean grade point average of 2.71 and a
standard deviation of 0.64. For an independent random sample of 123
students who did not vote, the mean grade point average was 2.79 and the
standard deviation was 0.56. Test, against a two-sided alternative, the null
hypothesis that the population means are equal.
7. Example 2
For a random sample of 125 British entrepreneurs, the mean number of job
changes was 1.91 and the sample standard deviation was 1.32. For an
independent random sample of 86 British corporate managers, the mean
number of job changes was 0.21 and the sample standard deviation was 0.53.
Test at 10%, the null hypothesis that the population means are equal against
the alternative that the mean number of job changes is higher for British
entrepreneurs than for British corporate managers.
8. Testing the Equality of Two Population Proportions (Large Samples)
• Now we develop procedures for comparing two population proportions. We consider a standard
model with a random sample of 𝑛𝑥 observations from a population with a proportion Px and a
second independent random sample of 𝑛𝑦 observations from a population with a proportion Py.
• Previously we learnt that for large samples, proportions can be approximated as normally
distributed random variables, and, as a result,
has a standard normal distribution. We want to test the hypothesis that the population proportions Px
and Py are equal.
9. • We want to test the hypothesis that the population proportions Px and
Py are equal. Denote their common value by 𝑃0. Then under this
hypothesis
H0 : Px - Py = 0 or H0 : Px = Py
follows to a close approximation a standard normal distribution.
10. • Finally, the unknown proportion 𝑃𝑜 can be estimated by a pooled estimator defined
as follows:
• The null hypothesis in these tests assumes that the population proportions are equal. If
the null hypothesis is true, then an unbiased and efficient estimator for 𝑃𝑜 can be obtained
by combining the two random samples, and, as a result, 𝑝0
is computed using this equation.
Then, we can replace the unknown 𝑃𝑜 by 𝑝0
to obtain a random variable that has a distribution close to the standard normal for large sample
sizes.
11. Example 3
A random sample of 1,556 people in country A were asked to respond to
this statement: Increased world trade can increase our per capita
prosperity. Of these sample members, 38.4% agreed with the statement.
When the same statement was presented to a random sample of 1,108
people in country B, 52.0% agreed. Test the null hypothesis that the
population proportions agreeing with this statement were the same in
the two countries against the alternative that a higher proportion agreed
in country B.
12. F-distribution
• There are a number of situations such as quality control, when we are interested in comparing the
variances from two normally distributed populations.
• Here we develop a procedure for testing the assumption that population variances from independent
samples are equal. To perform such tests, we introduce the F probability distribution.
• Suppose we take samples of sizes 𝑁1 and 𝑁2 respectively from either a single population or from two
populations with the same variance. We calculate the variance and then find the ratio. If we did this
for a large number of pairs of samples, the ratios form a distribution known as the F distribution
whose properties are known for various numbers of degrees of freedom.
• The F distribution is an asymmetric distribution that has a minimum value of 0, but no maximum
value. The curve reaches a peak not far to the right of 0, and then gradually approaches the horizontal
axis the larger the F value is.
13. • The f statistic, also known as an f value, is a random variable that has an F distribution.
𝐹 =
𝑆𝑥
2 /𝜎𝑥
2
𝑆𝑦
2 /𝜎𝑦
2
• The F distribution approaches, but never quite touches the horizontal axis.
• The F distribution has two degrees of freedom, 𝑉1 for the numerator, 𝑉2 for the denominator. For
each combination of these degrees of freedom there is a different F distribution.
• The F distribution is most spread out when the degrees of freedom are small. As the degrees of
freedom increase, the F distribution is less dispersed. f(4, 8) would refer to an F distribution
with v1 = 4 and v2 = 8 degrees of freedom
14.
15. Testing equality of variances of two normal populations:
In practical applications we usually arrange the F ratio so that the larger sample variance
is in the numerator and the smaller is in the denominator. Thus, we need to use only
the upper cutoff points to test the hypothesis of equality of variances. When the population
variances are equal, the F random variable becomes
𝐹 =
𝑆𝑥
2
𝑆𝑦
2
and this ratio of sample variances becomes the test statistic. The intuition for this test is
quite simple: If one of the sample variances greatly exceeds the other, then we must conclude
that the population variances are not equal. The hypothesis tests of equality of variances
are summarized as follows.
16.
17.
18. Example 4
It is hypothesized that the more expert a group of people examining personal income
tax filings, the more variable the judgments will be about the accuracy. Independent
random samples, each of 30 individuals, were chosen from groups with different levels
of expertise. The low-expertise group consisted of people who had just completed
their first intermediate accounting course.
Members of the high-expertise group had completed undergraduate studies and were
employed by reputable CPA firms. The sample members were asked to judge the
accuracy of personal income tax filings. For the
low-expertise group, the sample variance was 451.770, whereas for the high-expertise
group, it was 1,614.208. Test the null hypothesis that the two population variances are
equal against the alternative that the true variance is higher for the high-expertise
group.
