# Step-by-Step Guide:

# One Sample Proportion Tests

## Overview

Here you will learn the following:

how to run a One Sample Proportion Test (Binomial test - 2 outcomes);

how to run a One Sample Proportion Test (Chi-Square Goodness of Fit - multiple outcomes).

Both tests require categorical variables. These tests are very useful if you want to check whether your sample is similar to the population you are sampling.

Often you may know information about your population e.g. how gender or ethnicity are distributed. This test allows you to check against these known parameters.

## Dataset used for One Sample Proportion Tests:

Skoczylis, Joshua, 2021, "Extremism, Life Experiences and the Internet", https://doi.org/10.7910/DVN/ICTI8T, Harvard Dataverse, Version 3.

## Binomial Test (2 Outcomes): Step-by-Step Guide

## Binomial Test Hypothesis

Ho: The Gender variable has the same proportions as the UK population (there is no difference)

Ha: The Gender variable does not have the same proportions as the UK population (there is a difference)

## Binomial Test variables required

Variables should have two outcomes only (e.g. male/female or fail/pass)

Variable used in Example: Gender

## Binomial Test: Step-by-Step Guide

1.

Generate a descriptive table/plot

In the UK population, we have an almost equal split between males (49.1%) and females (50.9%) in the UK. Let's check how similar our sample is using descriptive statistics

Navigate to Analyses > Descriptives

Drag Gender into your variables box. Now select Frequency Distribution

If you prefer plots, you can also easily generate this in the Plot section. Just select a Bar Plot.

The results indicate that our sample is not equally distributed. In the next step we will check, whether this difference is statistically significant or not.

2.

Run your Binomial Test

Navigate to Analyses > Frequencies > 2 Outcome Binomial Test

Now drag the Gender variable into the Variable section

Adjust the test value where necessary. Let's say there were more women e.g. 60%, then you could adjust the test to 0.6.

It is always good to add the confidence interval to your output too.

Note: If you have a directional Hypothesis e.g. smaller than or larger than, then change this setting too.

A table with the results will appear on your right.

## Results: Reject/Accept Hypothesis

Reject Null Hypothesis

The descriptive plots and table above should have already given you a good indication that the distribution of Gender is different to the UK population. This test confirms this. The proportion of both males and females differs - hence we get a significant p-value for both.

Note: The p-values can differ for both outcomes, as one outcome may not be significantly different from the proportions you set.

We can therefore reject the Null Hypothesis (p <.001) in favour of the Alternative Hypothesis which states that the proportion of Males and Females in the UK population are different to the proportion in the sample.

## Chi-Square Goodness of Fit (Multiple Outcomes): Step-by-Step Guide

## Chi-Square Goodness of Fit Hypothesis

Ho: The Ethnicity variable has the same proportion as the UK population (no difference)

Ha: The Ethnicity variable does not have the same proportion as the UK population (there is a difference)

## Chi-Square Goodness of Fit Variables required

or

Variable should have more than two outcomes

Variable used in Example: Ethnicity

## Chi-Square Goodness of Fit: Step-by-Step Guide

Using a Proportion (Chi-Square Goodness of Fit) Test

by Datalabcc

1.

Generate a descriptive table/plot

The Office of National Statistics provides lots of information about the population. From the Census we have a pretty good idea of how ethnicity is distributed. Here we will check whether the extremism dataset has a similar distribution.

Navigate to Analyses > Descriptives

Drag Ethnicity into your variables box. Now select Frequency Distribution

If you prefer plots, you can also easily generate this in the Plot section (although your plot here will look better using Survey Plot or JJStats). Given the low percentages of non-white ethnicities, a table will give you the most accurate information.

The below table give us an accurate breakdown of the ethincity of our participants. In the next step we will compare this to the data from the ONS to see if there is a difference.

2.

Run your Proportion Test

Navigate to Analyses > Frequencies > N Outcomes Chi-Square Goodness of Fit

Now drag the Ethnicity variable into the Variables field.

In the Expected Proportion Section, you will need to enter the correct ratios for each of your variable outcomes

For our example variable, we need to enter the following data: Whites 17.2 (86%); Asian 1.5 (7.5%); Black 0.66 (3.35); Mixed 0.44 (2.2%); Other 0.2 (1%).

Note, your Proportions will not be correct until you have entered all the ratios.

If you don't have the rations (e.g. you might only have the percentages) you can work them out as follows:

Percentage/number of outcomes = Ratio

Example: For Whites it would be 86/5 = 17.2

## Results: Reject/Accept Hypothesis

Reject Null Hypothesis

Based on the test results (p <.001) we reject the Null Hypothesis and accepted the Alternative Hypothesis as there is a difference.

The table below includes the expected counts, to give you an idea of the difference.

We can say the ethnicity in our sample is not reflective of the ethincity within the wider population.