Statistic 6

In this document we will discuss 2 – sample T- Paired or Matched hypothesis

testing and confidence intervals that used a mean and sample SD.

There are still 3 different hypothesis scenarios with a 2 – Sample Paired

Hypothesis Test.

Lower Tail Test (1 tail):

Ho: �̅�𝑑𝑖𝑓𝑓 = 0

Ha: �̅�𝑑𝑖𝑓𝑓 < 0

Upper Tailed Test (1 tail):

Ho: �̅�𝑑𝑖𝑓𝑓 = 0

Ha: �̅�𝑑𝑖𝑓𝑓 > 0

Two Tailed Test:

Ho: �̅�𝑑𝑖𝑓𝑓 = 0

Ha: �̅�𝑑𝑖𝑓𝑓 ≠ 0

The hypothesized value is 0 and the same key words apply from a 1 – sample

hypothesis test to determine which scenario to use. �̅�𝑑𝑖𝑓𝑓 is the average of the

difference column. Paired samples are samples that share something in common.

They are dependent on one another. Since they share something in common the

samples go from 2 to 1. This will be very similar to the 1-sample T hypothesis test

in the discussion forum.

The T – Test Statistic = �̅�𝑑𝑖𝑓𝑓−0

𝑆𝐷𝑑𝑖𝑓𝑓

√𝑛

Where 𝑆𝐷𝑑𝑖𝑓𝑓 this is SD of the difference column

We can use =T.DIST, =T.DIST.RT and =T.DIST.2T to find the p-values. These should

look familiar from the discussion forum.

Example:

Blood plasma cancer is characterized by increased blood vessel formulation in the

bone marrow that is a predictive factor in survival. One treatment approach used

for blood plasma cancer is stem cell transplantation with the patient’s own stem

cells. Measurements were taken immediately prior to the stem cell transplant

and at the time complete response was determined. The estimate of the mean

difference, in bone marrow microvessel density before and after the stem cell

transplant. There are 7 patients that are sampled. Is the blood vessel formulation

in the bone marrow different after the stem cell transplant? Use alpha = .05.

Before After

187 168 199 183

175 155 177 160

193 180 188 175

179 166

Here we see that these sample “share” something in common because it is the

same patient and measurements were taken before and after treatment.

First step is to state the hypothesis scenario. Because the key word says different

this means it is a two tailed test.

Ho: �̅�𝑑𝑖𝑓𝑓 = 0

Ha: �̅�𝑑𝑖𝑓𝑓 ≠ 0

Before we start calculating anything by hand and because we are given the raw

data set, we can actually run this hypothesis test in Excel. And since you installed

the Data Analysis Toolpak it is easy to do.

Go to Data -> Data Analysis -> and scroll to where it says t-Test Paired Two

Samples for Means and click OK

Under Input:

Variable 1 Range: you will highlight the Before column and make sure you include

the top row where the Label is located.

Variable 2 Range: you will highlight the After column and make sure you include

the top row where the Label is located.

Check the “Labels” box because we did include the first row of labels. For Alpha

out 0.05 but this can be change depending on what significance level you use.

Then make sure the bubble for New Workbook Ply: highlight and click OK. It

should look similar to the screenshot below.

Once you click OK in a new Worksheet this should populate.

t-Test: Paired Two Sample for Means

Before After Mean 185.4285714 169.5714286

Variance 78.61904762 106.2857143 Observations 7 7

Pearson Correlation 0.963199569

Hypothesized Mean Difference

0

df 6

t Stat 14.13506279

P(T<=t) one-tail 3.91511E-06

t Critical one-tail 1.943180281

P(T<=t) two-tail 7.83022E-06

t Critical two-tail 2.446911851 Here we have all the values we need to state a conclusion.

We see the T- Test Statistic = 14.135 and because we ran a two tailed test the

p-value = 7.83022E-06. The “E-06” means scientific notation. We can rewrite this

as 7.83 x 10-6, then to convert this to decimal form we need to move the decimal

6 places to the left.

p -value = .00000783 < .05. This p-value is less than .05 which means we Reject

Ho. Yes, there is statistical evidence that blood vessel formulation in the bone

marrow different after the stem cell transplant.

If we were running a 1-tailed test, we are given the p-value which is 3.91511E-06.

Converting this to a decimal we get .000003915. The T-Test Statistic is the same

and so is the conclusion for a 1-tailed test.

Now that we ran a hypothesis test, let calculate a confidence interval and draw

the same conclusion.

The equation for a 2 – paired confidence interval:

�̅�𝑑𝑖𝑓𝑓 ±𝑇𝛼 2

∗ ∗ 𝑆𝐷𝑑𝑖𝑓𝑓

√𝑛

Where Standard Error (SE) = 𝑆𝐷𝑑𝑖𝑓𝑓

√𝑛

Margin of Error (ME) = 𝑇𝛼 2

∗ ∗ 𝑆𝐷𝑑𝑖𝑓𝑓

√𝑛

Before we can start plugging values into our equation we need to find the

Difference Column. We will subtract Before – After to get the Difference Column

and then calculate the average and SD based on that column.

Before After Diff Column 187 168 19

199 183 16

175 155 20 177 160 17

193 180 13

188 175 13

179 166 13

Average 15.85714286 SD 2.968084199

Plugging in what we know:

15.85714 ±𝑇𝛼 2

∗ ∗ 2.96808

√7

The last thing we need to find is the T- Critical Value. We will use the =T.INV.2T

function to find this. This function should look familiar from Week 4.

If we want to find a 95% confidence interval, then alpha = .05. We will use this

value in our Excel function and the DF is n – 1 = 7 – 1 = 6.

=T.INV.2T(.05,6)

We see the T – Critical Value is 2.44691. We will plug this into the equation and

solve. BUT if we look at our output when we ran the Test, the T-Critical Value was

also given to us as well.

t-Test: Paired Two Sample for Means

Before After Mean 185.4285714 169.5714286

Variance 78.61904762 106.2857143 Observations 7 7

Pearson Correlation 0.963199569

Hypothesized Mean Difference

0

df 6

t Stat 14.13506279

P(T<=t) one-tail 3.91511E-06

t Critical one-tail 1.943180281

P(T<=t) two-tail 7.83022E-06

t Critical two-tail 2.446911851

15.85714 ±𝑇𝛼 2

∗ ∗ 2.96808

√7

15.85714 ±2.446911∗ 2.96808

√7

15.85714 ±2.446911∗ 1.121828

15.85714 ±2.74501

(13.112, 18.60215)

The confidence interval goes from 13.112 to 18.60215. This interval goes from a

positive value to a positive value. This means that 0 is NOT in this interval.

Because 0 is NOT in the interval it is Significant, and we Reject Ho. This is the

same conclusion that we got with the hypothesis test.