# How to do a hypothesis test for the difference between two proportions

## Description

When dealing with qualitative data, we typically measure what proportion of the population falls into various categories (e.g., which religion a survey respondent adheres to, if any). We might want to compare two proportions by measuring their difference, and asking whether it is equal, greater, or less than zero. How can we perform such a test?

Related tasks:

- How to compute a confidence interval for the difference between two proportions
- How to do a hypothesis test for a mean difference (matched pairs)
- How to do a hypothesis test for a population proportion
- How to do a hypothesis test for population variance
- How to do a hypothesis test for the difference between means when both population variances are known
- How to do a hypothesis test for the mean with known standard deviation
- How to do a hypothesis test for the ratio of two population variances
- How to do a hypothesis test of a coefficient’s significance
- How to do a one-sided hypothesis test for two sample means
- How to do a two-sided hypothesis test for a sample mean
- How to do a two-sided hypothesis test for two sample means

## Using SciPy, in Python

We will use some fake data in this example, but you can replace it with your real data. Imagine we conduct a survey of people in Boston and of people in Nashville and ask them if they prefer chocolate or vanilla ice cream. We get data like the following.

City | Prefer chocolate | Prefer vanilla | Total |
---|---|---|---|

Boston | 60 | 90 | 150 |

Nashville | 85 | 50 | 135 |

We want to compare the proportions of people from the two cities who like vanilla.

Let $\bar{p}_1$ represent the proportion of people from Boston who like vanilla and $\bar{p}_2$ represent the proportion of people from Nashville who like vanilla.

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n1 = 150 # number of observations in sample 1
n2 = 135 # number of observations in sample 2
p_bar1 = 90/150 # proportion in sample 1
p_bar2 = 50/135 # proportion in sample 2

We choose a value $0 \le \alpha \le 1$ as our Type 1 error rate. For this example, we will use $\alpha=0.05$.

### Two-tailed test

In a two-tailed test, the null hypothesis states that the difference between the two proportions equals a hypothesized value; let’s choose zero, $H_0: \bar{p}_1 - \bar{p}_2 = 0$. We perform this test by computing a test statistic and $p$-value as shown below, then comparing the $p$-value to our chosen $\alpha$.

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import numpy as np
p_bar = (90 + 50) / (150 + 135) # overall proportion
std_error = np.sqrt(p_bar*(1-p_bar)*(1/n1+1/n2)) # standard error
test_statistic = (p_bar1 - p_bar2)/std_error # test statistic
from scipy import stats
2*stats.norm.sf(abs(test_statistic)) # two-tailed p-value

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0.00010802693662804402

Our $p$-value, 0.000108, is smaller than $\alpha$, so we can reject the null hypothesis and conclude that the difference between the two proportions is different from zero.

But we did not need to compare the difference to zero; we could have used any hypothesized difference for comparison. Let’s repeat the above test, comparing the difference to $0.15$ instead, as an example.

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import numpy as np
hyp_diff = 0.15 # hypothesized difference
std_error = np.sqrt(p_bar1*(1-p_bar1)/n1
+ p_bar2*(1-p_bar2)/n2) # standard error
test_statistic = ((p_bar1 - p_bar2) - hyp_diff)/std_error # test statistic
from scipy import stats
2*stats.norm.sf(abs(test_statistic)) # two-tailed p-value

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0.16744531573658772

Our $p$-value, 0.1674, is greater than $\alpha$, so we cannot reject the null hypothesis and cannot conclude that the difference between these two proportions is significantly different from 0.15.

### Right-tailed test

In a right-tailed test, the null hypothesis states that the difference between the two proportions is less than or equal to a hypothesized value. Let’s begin by using zero as our hypothesized value, $H_0: \bar{p}_1 - \bar{p}_2 \le 0$.

We repeat some code below that we’ve seen above, just to make it easy to copy and paste the example elsewhere.

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import numpy as np
p_bar = (90 + 50) / (150 + 135) # overall proportion
std_error = np.sqrt(p_bar*(1-p_bar)*(1/n1+1/n2)) # standard error
test_statistic = (p_bar1 - p_bar2)/std_error # test statistic
from scipy import stats
stats.norm.sf(abs(test_statistic)) # right-tailed p-value

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5.401346831402201e-05

Our $p$-value is smaller than $\alpha$, so we can reject the null hypothesis and conclude that the difference between the two proportions is significantly greater than zero.

But we did not need to compare the difference to zero; we could have used any hypothesized difference for comparison. Let’s repeat the above test, comparing the difference to $0.15$ instead, as an example.

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import numpy as np
hyp_diff = 0.15 # hypothesized difference
std_error = np.sqrt(p_bar1*(1-p_bar1)/n1
+ p_bar2*(1-p_bar2)/n2) # standard error
test_statistic = ((p_bar1 - p_bar2) - hyp_diff)/std_error # test statistic
from scipy import stats
stats.norm.sf(abs(test_statistic)) # right-tailed p-value

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0.08372265786829386

Our $p$-value, 0.0837, is greater than $\alpha$, so we cannot reject the null hypothesis and cannot conclude that the difference between these two proportions is significantly greater than 0.15.

### Left-tailed test

In a left-tailed test, the null hypothesis states that the difference between the two proportions is greater than or equal to a hypothesized value. Let’s begin by using zero as our hypothesized value, $H_0: \bar{p}_1 - \bar{p}_2 \ge 0$.

We repeat some code below that we’ve seen above, just to make it easy to copy and paste the example elsewhere.

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import numpy as np
p_bar = (90 + 50) / (150 + 135) # overall proportion
std_error = np.sqrt(p_bar*(1-p_bar)*(1/n1+1/n2)) # standard error
test_statistic = (p_bar1 - p_bar2)/std_error # test statistic
from scipy import stats
stats.norm.sf(-test_statistic) # left-tailed p-value

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0.999945986531686

Our $p$-value, 0.9999, is greater than $\alpha$, so we cannot reject the null hypothesis and cannot conclude that the difference between the two proportions is significantly less than zero.

But we did not need to compare the difference to zero; we could have used any hypothesized difference for comparison. Let’s repeat the above test, comparing the difference to $0.15$ instead, as an example.

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import numpy as np
hyp_diff = 0.15 # hypothesized difference
std_error = np.sqrt(p_bar1*(1-p_bar1)/n1
+ p_bar2*(1-p_bar2)/n2) # standard error
test_statistic = ((p_bar1 - p_bar2) - hyp_diff)/std_error # test statistic
from scipy import stats
stats.norm.sf(-test_statistic) # left-tailed p-value

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0.9162773421317061

Our $p$-value, 0.91627, is greater than $\alpha$, so we cannot reject the null hypothesis and cannot conclude that the difference between these two proportions is significantly less than 0.15.

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

## Solution, in R

We will use some fake data in this example, but you can replace it with your real data. Imagine we conduct a survey of people in Boston and of people in Nashville and ask them if they prefer chocolate or vanilla ice cream. We get data like the following.

City | Prefer chocolate | Prefer vanilla | Total |
---|---|---|---|

Boston | 60 | 90 | 150 |

Nashville | 85 | 50 | 135 |

We want to compare the proportions of people from the two cities who like vanilla.

Let $\bar{p}_1$ represent the proportion of people from Boston who like vanilla and $\bar{p}_2$ represent the proportion of people from Nashville who like vanilla.

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n1 <- 150
n2 <- 135
p_bar1 <- 90/150
p_bar2 <- 50/135

We choose a value $0 \le \alpha \le 1$ as our Type 1 error rate. For this example, we will use $\alpha=0.05$.

### Two-tailed test

In a two-tailed test, the null hypothesis states that the difference between the two proportions equals a hypothesized value; let’s choose zero, $H_0: \bar{p}_1 - \bar{p}_2 = 0$. We perform this test by computing a test statistic and $p$-value as shown below, then comparing the $p$-value to our chosen $\alpha$.

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p_bar <- (90 + 50) / (150 + 135) # overall proportion
std_error <- sqrt(p_bar*(1-p_bar)*(1/n1+1/n2)) # standard error
test_statistic <- (p_bar1 - p_bar2)/std_error # test statistic
2*pnorm(q = test_statistic, lower.tail = FALSE) # two-tailed p-value

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[1] 0.0001080269

Our $p$-value, 0.000108, is smaller than $\alpha$, so we can reject the null hypothesis and conclude that the difference between the two proportions is different from zero.

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hyp.diff = 0.15 # hypothesized difference
std_error <- sqrt(p_bar1*(1-p_bar1)/n1
+ p_bar2*(1-p_bar2)/n2) # standard error
test_statistic <- ((p_bar1 - p_bar2) - hyp.diff)/std_error # test statistic
2*pnorm(q = test_statistic, lower.tail = FALSE) # two-tailed p-value

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[1] 0.1674453

Our $p$-value, 0.1674, is greater than $\alpha$, so we cannot reject the null hypothesis and cannot conclude that the difference between these two proportions is significantly different from 0.15.

### Right-tailed test

In a right-tailed test, the null hypothesis states that the difference between the two proportions is less than or equal to a hypothesized value. Let’s begin by using zero as our hypothesized value, $H_0: \bar{p}_1 - \bar{p}_2 \le 0$.

We repeat some code below that we’ve seen above, just to make it easy to copy and paste the example elsewhere.

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p_bar <- (90 + 50) / (150 + 135) # overall proportion
std_error <- sqrt(p_bar*(1-p_bar)*(1/n1+1/n2)) # standard error
test_statistic <- (p_bar1 - p_bar2)/std_error # test statistic
pnorm(q = test_statistic, lower.tail = FALSE) # right-tailed p-value

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[1] 5.401347e-05

Our $p$-value is smaller than $\alpha$, so we can reject the null hypothesis and conclude that the difference between the two proportions is significantly greater than zero.

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hyp.diff = 0.15 # hypothesized difference
std_error <- sqrt(p_bar1*(1-p_bar1)/n1
+ p_bar2*(1-p_bar2)/n2) # standard error
test_statistic <- ((p_bar1 - p_bar2) - hyp.diff)/std_error # test statistic
pnorm(q = test_statistic, lower.tail = FALSE) # right-tailed p-value

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[1] 0.08372266

Our $p$-value, 0.0837, is greater than $\alpha$, so we cannot reject the null hypothesis and cannot conclude that the difference between these two proportions is significantly greater than 0.15.

### Left-tailed test

In a left-tailed test, the null hypothesis states that the difference between the two proportions is greater than or equal to a hypothesized value. Let’s begin by using zero as our hypothesized value, $H_0: \bar{p}_1 - \bar{p}_2 \ge 0$.

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p_bar <- (90 + 50) / (150 + 135) # overall proportion
std_error <- sqrt(p_bar*(1-p_bar)*(1/n1+1/n2)) # standard error
test_statistic <- (p_bar1 - p_bar2)/std_error # test statistic
pnorm(q = test_statistic, lower.tail = TRUE) # left-tailed p-value

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[1] 0.999946

Our $p$-value, 0.9999, is greater than $\alpha$, so we cannot reject the null hypothesis and cannot conclude that the difference between the two proportions is significantly less than zero.

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hyp.diff = 0.15 # hypothesized difference
std_error <- sqrt(p_bar1*(1-p_bar1)/n1
+ p_bar2*(1-p_bar2)/n2) # standard error
test_statistic <- ((p_bar1 - p_bar2) - hyp.diff)/std_error # test statistic
pnorm(q = test_statistic, lower.tail = TRUE) # left-tailed p-value

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[1] 0.9162773

Our $p$-value, 0.91627, is greater than $\alpha$, so we cannot reject the null hypothesis and cannot conclude that the difference between these two proportions is significantly less than 0.15.

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

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