Limited Time Offer: Save 10% on all 2022 Premium Study Packages with promo code: BLOG10 # Difference Between Two Population Means

It is common for analysts to establish whether there is a significant difference between the means of two different populations. For instance, they might want to know whether the average returns for two subsidiaries of a given company exhibit a significant difference. The results of such a test may then inform decisions regarding resource allocation or the rewarding of directors. Before embarking on such an exercise, it is paramount to ensure that the samples taken are independent and sourced from normally distributed populations.

When testing for the difference between two population means, we always use the student’s t-distribution. Thus, we can subdivide the tests for the difference between means into two distinctive scenarios.

## Difference Between Two Population Means: Small Samples With a Common (Pooled) Variance

Basic situation: two independent random samples of sizes n1 and n2, means X’1 and X’2, and variances $$\sigma_1^2$$ and $$\sigma_1^2$$ respectively. We assume that $$\sigma_1^2 = \sigma_1^2 = \sigma^2$$

H0:  μ1 – μ2 = 0

We estimate the common variance for the two samples by $$S_p^2$$ where,

$${ S }_{ p }^{ 2 }=\frac { \left( { n }_{ 1 }-1 \right) { S }_{ 1 }^{ 2 }+\left( { n }_{ 2 }-1 \right) { S }_{ 2 }^{ 2 } }{ { n }_{ 1 }+{ n }_{ 2 }-2 }$$

We use the t-statistic with (n1 + n2 – 2) degrees of freedom, under the null hypothesis that μ1 – μ2 = 0. Therefore,

$${ t }_{ { n }_{ 1 }+{ n }_{ 2 }-2 }=\frac { { \bar { x } }_{ 1 }-{ \bar { x } }_{ 2 } }{ { S }_{ p }\sqrt { \left( \frac { 1 }{ { n }_{ 1 } } +\frac { 1 }{ { n }_{ 2 } } \right) } }$$

## Difference Between Two Population Means: Large Samples

Basic situation: two independent random samples of sizes n1 and n2, means X’1 and X’2, and Unknown variances $$\sigma_1^2$$ and $$\sigma_1^2$$ respectively.

H0:  μ1 – μ2 = 0

When dealing with large samples, we can use S2 to estimate σ2. This is made possible by the central limit theorem. As such, the requirement to draw a sample from a normally distributed population is not necessary. The test statistic used is:

$$Z=\frac { { \bar { x } }_{ 1 }-{ \bar { x } }_{ 2 } }{ \sqrt { \left( \frac { { \sigma }_{ 1 }^{ 2 } }{ { n }_{ 1 } } +\frac { { \sigma }_{ 2 }^{ 2 } }{ { n }_{ 2 } } \right) } }$$

We then compare the test statistic with the relevant percentage point of the normal distribution. The test statistic is also applicable when the variances are known.

### Example 1: Small Samples Assumed to Have a Common Variance

Nutritional experts want to establish whether obese patients on a new special diet have a lower weight than the control group. After 6 weeks, the average weight of 10 patients (group A) on the special diet is 75kg, while that of 10 more patients of the control group (B) is 72kg. Carry out a 5% test to determine if the patients on the special diet have a lower weight.

Additional information: $$\sum A^2 = 59520$$ and $$\sum B^2 =56430$$

Solution

As is the norm, start by stating the hypothesis:

H0: μB – μA = 0

H1: μB > μA

We assume that the two samples have equal variance, are independent and distributed normally. Then, under the H0,

$$\frac { \bar { B } -\bar { A } }{ S\sqrt { \frac { 1 }{ m } +\frac { 1 }{ n } } } \sim { t }_{ m+n-2 }$$

\begin{align*} { S }_{ A }^{ 2 } & =\frac { \left\{ 59520-{ \left( 10\ast { 75 }^{ 2 } \right) } \right\} }{ 9 } =363.33 \\ { S }_{ B }^{ 2 } & =\frac { \left\{ 56430-{ \left( 10\ast { 72}^{ 2 } \right) } \right\} }{ 9 } =510 \\ \end{align*}

Therefore,

$$S^p_2 =\cfrac {(9 * 363.33 + 9 * 510)}{(10 + 10 -2)} = 436.665$$

And

$$\text{the test statistic} =\cfrac {(75 -72)}{ \left\{ \sqrt{439.665} * \sqrt{ \left(\frac {1}{10} + \frac {1}{10}\right)} \right\} }= 0.3210$$

Our test statistic (0.3210) is less than the upper 5% point (1. 734) of the t-distribution with 18 degrees of freedom. Therefore, we do not have sufficient evidence to reject the H0 at 5% significance. As such, it is reasonable to conclude that the special diet has the same effect on body weight as the placebo.

Note: You could choose to work with the p-value and determine P(t18 > 0.937) and then establish whether this probability is less than 0.05. However, working out the problem correctly would lead to the same conclusion as above.

### Example 2: Small Samples Assumed to Have a Common Variance

Suppose we replace ‘>’ with ‘≠’ in H1 in the example above, would the decision rule change?

Replacing ‘>’ with ‘≠’ in H1 would change the test from a one-tailed one to a two-tailed test. We would compute the test statistic just as demonstrated above. However, we would have to divide the level of significance by 2 and compare the test statistic to both the lower and upper 2.5% points of the t18 -distribution (±2.101). Our test statistic lies within these limits (non-rejection region). The decision rule would, therefore, remain unchanged.

Shop CFA® Exam Prep

Offered by AnalystPrep Level I
Level II
Level III
All Three Levels
Featured Shop FRM® Exam Prep FRM Part I
FRM Part II
FRM Part I & Part II
Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success
Shop Actuarial Exams Prep Exam P (Probability)
Exam FM (Financial Mathematics)
Exams P & FM
Shop GMAT® Exam Prep Complete Course Sergio Torrico
2021-07-23
Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar. diana
2021-07-17
So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings. Kriti Dhawan
2021-07-16
A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep. nikhil kumar
2021-06-28
Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures. Marwan
2021-06-22
Great support throughout the course by the team, did not feel neglected Benjamin anonymous
2021-05-10
I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way! michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.