Testing for a Difference in Means
We can perform a formal hypothesis test to answer questions concerning a difference in two population means.
We’ll use the same steps introduced in our Introduction to Hypothesis Testing notes, but some of the details will differ in this new scenario.
1. State the question of interest
Motivating Example:
Is there a difference in the average number of math courses taken between graduates at University A and University B?
2. Identify the parameter of interest
Motivating Example:
The difference in the average number of math courses taken, \(\mu_A - \mu_B\)
3. State the null and alternative hypotheses
Null hypothesis
\[H_0: \mu_1 = \mu_2\]
Alternative hypothesis:
This depends on the question of interest.
Lower one-sided
Question of interest: Is the mean of population 1 less than the mean of population 2?
\(H_A: \mu_1 <\mu_2\)
Upper one-sided
Question of interest: Is the mean of population 1 greater than the mean of population 2?
\(H_A: \mu_1 >\mu_2\)
Two-sided
Question of interest: Is the mean of population 1 different from the mean of population 2?
\(H_A: \mu_1 \neq\mu_2\)
3. State the null and alternative hypotheses
Motivating Example:
\(H_0:\)\(\mu_A = \mu_B\)
\(H_A:\mu_A \neq \mu_B\)
4. Using the sampled data and the alternative hypothesis, determine what values would be considered “as or more extreme” than the observed sampled statistic.
Lower one-sided: \(H_A: \mu_1<\mu_2\)
Upper one-sided: \(H_A: \mu_1 >\mu _2\)
Two-sided: \(H_A: \mu_1 \neq \mu_2\)
4. Using the sampled data and the alternative hypothesis, determine what values would be considered “as or more extreme” than the observed sampled statistic.
Motivating Example:
\(\overline{x}_A - \overline{x}_B = 4 - 3.5 = 0.5\)
Any difference in sample means greater than 0.5 in magnitude would be considered as or more extreme than the observed sampled statistic.
5. Determine the Null Distribution
If the sample sizes are sufficiently large, under the null hypothesis, the distribution of the test statistic used in testing the difference between two population means is a
t distribution with \(\nu\) degrees of freedom.
\(\nu\) represents the Satterthwaite degrees of freedom.
How large do the samples need to be?
If both \(n_1\geq 30\) and \(n_2 \geq 30\), we can move forward.
If either \(n_1 < 30\) or \(n_2 < 30\), we need to look at the sampled distribution(s) of the small sample(s). If there are no clear outliers or strong skewness in the sampled data, we can move forward. If either sampled distribution suggests skewness or outliers, we should not proceed.
5. Determine the Null Distribution
Motivating Example:
Are the sample sizes sufficiently large?
Yes, both \(n_1\geq\) and \(n_2\geq30\).
So the null distribution is at distribution with 75.897 degrees of freedom.
6. Calculate the test statistic
When testing the difference in population means, \(\mu_1\) vs. \(\mu_2\), the test statistic is
\[t = \frac{\overline{x}_1-\overline{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]
Motivating Example:
\[t = \frac{4-3.5}{\sqrt{\frac{1.5^2}{46} + \frac{1^2}{32}}} = 1.766\]
7. Calculate the p-value using the test statistic and null distribution.
Lower one-sided:
\(H_A: \mu_1<\mu_2\)
Upper one-sided:
\(H_A: \mu_1>\mu_2\)
Two-sided:
\(H_A: \mu_1\neq \mu_2\)
R code:
2*(1-pt(abs(t), df))
where df is the Satterthwaite degrees of freedom, \(\nu\)
7. Calculate the p-value using the test statistic and null distribution.
Motivating Example:
2*(1-pt(abs(1.766), 75.897)) = 0.0814
8. Make a conclusion
Write a 4-part conclusion. The conclusion should be written in the context of the problem and contain the following components:
A statement for the strength of evidence in favor the alternative hypothesis.
Whether to reject or fail to reject the null hypothesis.
The point estimate for the parameter of interest.
A \((1-\alpha)100\%\) confidence interval estimate for the parameter of interest.
8. Make a conclusion
A statement in terms of the alternative hypothesis
- Using terms like “reject” and “fail to reject the null” may be confusing to novice readers.
- We’ll provide a more complete conclusion by providing a statement of evidence in terms of the alternative hypothesis that reflects the question of interest.
8. Make a conclusion
Motivating Example: Write a 4-part conclusion with a \(\alpha=0.01\) significance level.
1. There is slightly suggestive evidence that the graduates at University A and University B take a different number of math courses, on average.
2. At the \(\alpha=0.01\) significance level, we fail to reject the null hypothesis.
3. and 4. We are 99% confident that the students at University A take 0.248 fewer to 1.248 more math courses than students at University B on average, with an estimate difference in the average number of math courses of 0.5.
Practice! 🎩
Please locate the Class 15 Activity in Top Hat and answer Questions 1-7.
