12.6: Two Paired Means T Test
- Page ID
- 105871
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Next, we will learn how to apply the Two Paired Means \(T\) Procedure to test a statistical claim about 2 paired population means. Consider the following example.
A local group of economists wants to study the effects of the pandemic on the small businesses in their city. A survey was sent out to a random sample of 100 small businesses to anonymously report their weekly earnings throughout March and April. Only 10 businesses provided enough data to compute their average weekly earnings for two months:
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
|---|---|---|---|---|---|---|---|---|---|---|
|
March |
8.1 |
7.4 |
7.2 |
6.9 |
7.3 |
6.6 |
6.1 |
5.9 |
7.5 |
7.7 |
|
April |
4.8 |
2.4 |
3.2 |
4.1 |
1.9 |
2.1 |
4.6 |
2.6 |
1.6 |
4.2 |
Use \(5\%\) significance level to test the claim that the weekly earnings of small businesses have decreased during the pandemic.
Note that the sample means and the sample standard deviations for each sample can be easily computed. However, it appears that there is a clear pairing between the samples since it is the same 10 businesses for which the data was obtain in March and April, so we are not interested in the differences between the two samples but in the sample of differences.
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
Avg |
St. dev. |
|
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
March-April |
8.1-4.8 |
7.4-2.4 |
7.2-3.2 |
6.9-4.1 |
7.3-1.9 |
6.6-2.1 |
6.1-4.6 |
5.9-2.6 |
7.5-1.6 |
7.7-4.2 |
||
|
Difference |
3.3 |
5 |
4 |
2.8 |
5.4 |
4.5 |
1.5 |
3.3 |
5.9 |
3.5 |
3.92 |
1.32 |
As a result, we turned two samples into one with the sample mean \(\bar{x}_d=3.92\) thousand of dollars and sample standard deviation \(s_d=1.32\) thousand of dollars.
Now, let’s identify the statistical claim that needs to be tested:
“the weekly earnings of small businesses have decreased during the pandemic”
The claim is that the weekly earnings of small businesses in March is greater than the weekly earnings of small businesses in April, so we can symbolically express the claim as
\(\mu_1>\mu_2\) or \(\mu_1-\mu_2>0\) or \(\mu_d>0\)
Since the claim is about the paired means and the standard deviation of the population of differences is unknown, we will use the 1 Mean \(T\) Procedure.
Let’s check if all necessary assumptions are satisfied:
- The sample is simple random
- The population is normally distributed, or the samples size is greater than 30
- The population standard deviation, \(\sigma_d\), is unknown but \(s_d=1.32\)
We will use the following template to perform the hypothesis testing:
In step 1, we will set up the hypothesis:
Since the claim is in the form of an inequality, therefore it must be set as an alternative hypothesis, therefore the null hypothesis is \(\mu_d=0\) and the test is right-tail, that is:
|
\(H_0: \mu_d=0\) \(H_a: \mu_d>0\)
|
RT |
In step 2, we will identify the significance level:
The significance level can always be found in the text of the problem. In our case it is 5%, thus:
\(\alpha=0.05\)
In step 3, we will find the test statistic using the formula:
\(t_0=\frac{\bar{d}-\mu_0}{s_d/\sqrt{n}}=\frac{3.92-0}{1.32/\sqrt{10}}=9.391\)
In step 4, we will perform either the critical value approach or p-value approach to test the claim:
- In critical value approach, we construct the rejection region using the \(T\)-curve with \(df=n-1=10-1=9\):
RR: greater than \(t_{0.05}=1.833\)
- In p-value approach, we compute the p-value:
P-Value: \(P(T>9.391)=0.000\)
In step 5, we will draw the conclusion:
- In the critical value approach, we must check whether the test statistic is in the rejection region or not. Our test statistic is \(9.391\) and it is to the right of the critical value \(1.833\), thus it is in the rejection region.
- In the p-value approach, we must check whether the p-value is less than the significance level or not. Our p-value is \(0.000\) and it is less than \(\alpha=0.05\).
Both tests suggest that we DO reject the null hypothesis in favor of the alternative.
In step 6, we will interpret the results:
Under \(5\%\) significance level we have sufficient evidence to suggest that the weekly earnings of small businesses have decreased during the pandemic.
We discussed how to apply the Two Paired Means \(T\) Procedure to test a statistical claim about two paired population means with population standard deviations unknown.