Hypothesis Tests for Differences in Matched Pairs (College Board AP® Statistics)

Study Guide

Written by: Naomi C

Reviewed by: Dan Finlay

Paired t-test

What is a paired t-test?

A paired t-test is used to test whether or not the population means of two pieces of data that are linked are equal by examining the differences between paired data
A paired t-test is easily confused with a two-sample t-test
- but the data for a two-sample t-test must come from two independent populations
  - e.g. the mean lifespan of two different brands of battery
- whereas the data for a paired t-test is linked and comes from one population
  - e.g. the result of a blood test for a group of people before a medication is given and the result after it is given
To try to prove your case, you take two measures from a recent random sample of size $n$ from a single population and run a one-sample test on the differences between the two values
- e.g. you take a random sample of 30 students and record their test scores in both Science and Math, then run a one-sample t-test on the differences between each data pair

What are the hypotheses for a paired t-test?

The null hypothesis, $H_{0}$ , is the assumption that mean of the differences between the two measures is 0
- e.g. $H_{0} :$ The mean of the differences between the test score for Science and the test score for Math is 0, $μ_{d} = 0$
  - It is assumed to be correct, unless evidence proves otherwise

The alternative hypothesis, $H_{a}$ , is what you think the mean of the differences is relative to 0, i.e. if it is positive or negative
- e.g. $H_{a} :$ The mean of the differences between the test score for Science and the test score for Math is less than 0, $μ_{d} < 0$
- Remember that a t-test could be one-tailed or two-tailed, this will affect your alternative hypothesis

Examiner Tips and Tricks

When writing out your hypotheses, always fully define the symbol used for the population parameters in context, e.g. '... where $μ_{d}$ is the mean of the differences between the test score for Science and the test score for Math, and $d = x_{S c i e n c e} - x_{M a t h}$ '.

What are the conditions for a paired t-test?

When performing a paired t-test, you must show that it meets the following conditions:
- The two measures should come from the same items within the population
- Items in the sample (or experiment) must satisfy the independence condition
  - by verifying that data is collected by random sampling
  - or random assignment (in an experiment)
  - and, if sampling without replacement, showing that the sample size is less than 10% of the population size
- The distribution of the differences is approximately normally distributed
  - The shape needs to be approximately symmetric
  - There should be no outliers
- If the distribution of the differences is very skewed, you can only do a t-test if $n \geq 30$

How do I calculate the standardized test statistic (t-value)?

The t-value, for the paired t-test is given by:
- $t = \frac{{\bar{x}}_{d} - μ_{d}}{\frac{s_{d}}{\sqrt{n}}}$
- where ${\bar{x}}_{d}$ is the mean of the difference between each pair of data in the sample, $s_{d}$ is the standard deviation of the differences in the sample, and $n$ is the sample size

Examiner Tips and Tricks

The formula for the standardized test statistic is given in the exam, $\frac{statistic - parameter}{standard error of the statistic}$ , along with tables of parameters and standard errors.

You will need to apply this correctly to get the t-value. In the numerator, the statistic is the mean of the pair differences in the sample and the parameter is the mean of the pair differences in the population, which is 0 for the null hypothesis.

How do I calculate the p-value?

Work out the t-value
Find the appropriate number of degrees of freedom ('dof')
- For a paired t-test, $dof = n - 1$
Using the t-distribution table given to you:
- find the row that corresponds to the dof
- identify the t-value in the row that is closest to the calculated value
- write down the value in the corresponding column header
  - this is the p-value
Note that the p-value from the t-table is for one tail

How do I conclude a hypothesis test?

Conclusions to a hypothesis test need to show two things:
- a decision about the null hypothesis
- an interpretation of this decision in the context of the question
To make the decision, compare the p-value to the significance level, $α$
- If $p < α$ then the null hypothesis should be rejected
- If $p > α$ then the null hypothesis should not be rejected
In a two-tailed test, double the p-value and compare this to $α$
- This is because there are two regions that are as extreme, or more extreme, than the one observed in the sample

Examiner Tips and Tricks

Remember that the test should be interpreted within the context of the question.

Use the same language in your conclusion that is used in the problem, e.g. 'The data provides sufficient evidence that the mean of the differences between the test score for Science and the test score for Math is less than 0, so students perform better in Math than Science'.

What are the steps for performing a paired t-test on a calculator?

When using a calculator to conduct a paired t-test, you must still write down all steps of the hypothesis testing process:
- State the null and alternative hypotheses and clearly define your parameters
- Describe the test being used and show that the situation meets the conditions required
- Calculate the t-value and the degrees of freedom
- Calculate the p-value using your calculator
  - select a one-sample t-test and enter the relevant summary statistics or data to generate the p-value for the differences
- Compare the p-value to the significance level
- Write down the conclusion to the test and interpret it in the context of the problem

Examiner Tips and Tricks

Even if you perform the paired t-test on your calculator, it is still important to show all of your working to demonstrate full understanding. Therefore you should still show workings for calculating the t-value and the degrees of freedom.

Worked Example

A research laboratory are conducting trials to investigate the effect of a new medication on blood pressure. It is believed that the medication will effectively lower blood pressure for people that currently have high blood pressure. 9 volunteers with a history of high blood pressure are selected at random from a pool of 100 and their systolic blood pressure is measured. They are then given a dose of the new medication and their systolic blood pressure is again measured 30 minutes after receiving it. You may assume that the differences in systolic blood pressure before and after the medication follow a normal distribution. The results of the trial are shown in the table below.

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)
1	156	151
2	162	158
3	149	147
4	168	165
5	153	152
6	165	161
7	159	156
8	151	151
9	164	158

Perform an appropriate hypothesis test at the 1% significance level. Is the medication effective at reducing high blood pressure?

Method 1: Using the t-table

State the type of test being used and verify the conditions for the test

The correct inference procedure is a paired t-test with $α = 0.01$

Both pieces of data were taken from each volunteer
The independence condition is satisfied, as
- the sample of 9 volunteers was selected at random
- and the sample size, 9, is less than 10% of the population, 100 (9 < 10)
It is assumed that the differences in systolic blood pressure before and after the medication follow a normal distribution
The sample size is small ( $n = 9$ , which is less than 30) and the population standard deviation is unknown, so the t-distribution can be used

Define the population parameter, $μ_{d}$

Let $μ_{d}$ be the mean of the differences between the initial blood pressure of a volunteer and their blood pressure after taking medication, where $d = blood pressure before - blood pressure after$

Write the null and alternative hypotheses

This will be a one-tailed test as it is believed that the medication will effectively lower blood pressure, if blood pressure is lowered then blood pressure before - blood pressure after will be greater than 0

$H_{0} : μ_{d} = 0 H_{a} : μ_{d} > 0$

Add an additional column to the table to summarize the differences of the blood pressure results

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)	Difference in blood pressure (mmgH)
1	156	151	5
2	162	158	4
3	149	147	2
4	168	165	3
5	153	152	1
6	165	161	4
7	159	156	3
8	151	151	0
9	164	158	6

Use your calculator, or work out by hand, the mean and the standard deviation of the differences

${\bar{x}}_{d} = 3.11111 . . . s_{d} = 1.90029 . . .$

Calculate the standardized test statistic

$\begin{array}{rcl} t & = & \frac{{\bar{x}}_{d} - μ_{d}}{\frac{s_{d}}{\sqrt{n}}} \\ = & \frac{3.11111 . . . - 0}{\frac{1.90029 . . .}{\sqrt{9}}} \\ = & 4.911 . . . \end{array}$

Calculate the number of degrees of freedom

degrees of freedom = 9 - 1 = 8

Find the p-value from the t-tables

Find the row corresponding to 8 degrees of freedom and identify the t-value that is closest to the absolute calculated t-value (4.911)

closest t-value = 5.041

corresponding p-value is $p = 0.0005$

Compare the probability to the significance level and state the conclusion of the test

$\begin{array}{rcl} 0.0005 & < & 0.01 \\ p & < & α \end{array}$

$H_{0}$ is rejected

Interpret the result in the context of the question

We have sufficient evidence to support the belief that the medication will reduce blood pressure in those that currently have high blood pressure

Method 2: Using a calculator

State the type of test being used and verify the conditions for the test

The correct inference procedure is a paired t-test with $α = 0.01$

Both pieces of data were taken from each volunteer
The independence condition is satisfied, as
- the sample of 9 volunteers was selected at random
- and the sample size, 9, is less than 10% of the population, 100 (9 < 10)
It is assumed that the differences in systolic blood pressure before and after the medication follow a normal distribution
The sample size is small ( $n = 9$ , which is less than 30) and the population standard deviation is unknown, so the t-distribution can be used

Define the population parameter, $μ_{d}$

Write the null and alternative hypotheses

$H_{0} : μ_{d} = 0 H_{a} : μ_{d} > 0$

Add an additional column to the table to summarize the differences of the blood pressure results

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)	Difference in blood pressure (mmgH)
1	156	151	5
2	162	158	4
3	149	147	2
4	168	165	3
5	153	152	1
6	165	161	4
7	159	156	3
8	151	151	0
9	164	158	6

Use your calculator, or work out by hand, the mean and the standard deviation of the differences

${\bar{x}}_{d} = 3.11111 . . . s_{d} = 1.90029 . . .$

Calculate the standardized test statistic

$\begin{array}{rcl} t & = & \frac{{\bar{x}}_{d} - μ_{d}}{\frac{s_{d}}{\sqrt{n}}} \\ = & \frac{3.11111 . . . - 0}{\frac{1.90029 . . .}{\sqrt{9}}} \\ = & 4.911 . . . \end{array}$

Calculate the number of degrees of freedom

degrees of freedom = 9 - 1 = 8

Create a table in your calculator of the difference in blood pressure column

Perform a one-sample t-test using the data, remember that the population mean of the differences, $μ_{d} = 0$ , and the alternative hypothesis, $μ_{d} < 0$

Calculate the $p$ -value

$p = 0.00058 . . .$

Compare the probability to the significance level and state the conclusion of the test

$\begin{array}{rcl} 0.0005 & < & 0.01 \\ p & < & α \end{array}$

$H_{0}$ is rejected

Interpret the result in the context of the question

We have sufficient evidence to support the belief that the medication will reduce blood pressure in those that currently have high blood pressure

Last updated: 24 October 2024

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Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)
1	156	151
2	162	158
3	149	147
4	168	165
5	153	152
6	165	161
7	159	156
8	151	151
9	164	158

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)	Difference in blood pressure (mmgH)
1	156	151	5
2	162	158	4
3	149	147	2
4	168	165	3
5	153	152	1
6	165	161	4
7	159	156	3
8	151	151	0
9	164	158	6

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)	Difference in blood pressure (mmgH)
1	156	151	5
2	162	158	4
3	149	147	2
4	168	165	3
5	153	152	1
6	165	161	4
7	159	156	3
8	151	151	0
9	164	158	6

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)
1	156	151
2	162	158
3	149	147
4	168	165
5	153	152
6	165	161
7	159	156
8	151	151
9	164	158

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)	Difference in blood pressure (mmgH)
1	156	151	5
2	162	158	4
3	149	147	2
4	168	165	3
5	153	152	1
6	165	161	4
7	159	156	3
8	151	151	0
9	164	158	6

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)	Difference in blood pressure (mmgH)
1	156	151	5
2	162	158	4
3	149	147	2
4	168	165	3
5	153	152	1
6	165	161	4
7	159	156	3
8	151	151	0
9	164	158	6

Hypothesis Tests for Differences in Matched Pairs (College Board AP® Statistics)

Study Guide

Paired t-test

What is a paired t-test?

What are the hypotheses for a paired t-test?

What are the conditions for a paired t-test?

How do I calculate the standardized test statistic (t-value)?

How do I calculate the p-value?

How do I conclude a hypothesis test?

What are the steps for performing a paired t-test on a calculator?

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Unit 1: Exploring One-Variable Data

Summary Statistics

Describing Variables

Parameters & Statistics

Measures of Center

Measures of Position

Measures of Variability

Tables & Relative Frequency

Grouped Data

Outliers & Resistant Measures

Five-Number Summary & Boxplots

Skewness of Data

Comparing Data using Summary Statistics

Graphical Representations

Shape of Distributions

Bar Charts & Histograms

Dotplots & Stemplots

Cumulative Graphs

Comparing Univariate Graphs

The Normal Distribution

Properties of Normal Distributions

Standardized z-scores

Comparing Normal Distributions

Finding Proportions from Normal Distributions

Inverse Normal Calculations

Estimating Parameters of Normal Distributions

Unit 2: Exploring Two-Variable Data

Tables & Graphs

Two-Way Tables & Relative Frequencies

Bar Graphs & Mosaic Plots

Scatterplots & Regression

Explanatory & Response Variables

Scatterplots

Association & Correlation Coefficients

Interpolation & Extrapolation using Linear Models

Residuals

The Least-Squares Regression Line

Residual Plots

The Coefficient of Determination

Outliers, High-Leverage & Influential Points

Linearization of Bivariate Data

Unit 3: Collecting Data

Sampling Methods & Bias

Introduction to Sampling

Simple Random Sampling (SRS)

Random Sampling Methods

Types of Bias

Non-random (Biased) Sampling Methods

Experimental Design

Introduction to Experiments

Well-Designed Experiments

Control Groups, Placebos & Blind Experiments

Completely Randomized Design

Randomized Block & Matched Pairs Design

Unit 4: Probability, Random Variables & Probability Distributions

Probability

Estimating Probability using Relative Frequency

Probabilities of Single Events

Introduction to Combined Events

Addition Rule & Mutually Exclusive Events

Conditional Probability

Multiplication Rule & Independent Events

Probabilities of Combined Events using Tree Diagrams

Probabilities of Combined Events using the Rules

Discrete Random Variables

Probability Distributions for Discrete Random Variables

Cumulative Probability Distributions for Discrete Random Variables

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)
1	156	151
2	162	158
3	149	147
4	168	165
5	153	152
6	165	161
7	159	156
8	151	151
9	164	158

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)	Difference in blood pressure (mmgH)
1	156	151	5
2	162	158	4
3	149	147	2
4	168	165	3
5	153	152	1
6	165	161	4
7	159	156	3
8	151	151	0
9	164	158	6

Volunteer	Systolic blood pressure before (mmgH)	Systolic blood pressure after (mmgH)	Difference in blood pressure (mmgH)
1	156	151	5
2	162	158	4
3	149	147	2
4	168	165	3
5	153	152	1
6	165	161	4
7	159	156	3
8	151	151	0
9	164	158	6