z test for the difference between two proportions overview

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$z$ test for the difference between two proportions	Sign test	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Independent/grouping variable	Independent variable
One categorical with 2 independent groups	2 paired groups
Dependent variable	Dependent variable
One categorical with 2 independent groups	One of ordinal level
Null hypothesis	Null hypothesis
H₀: $\pi_1 = \pi_2$ Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2.	H₀: P(first score of a pair exceeds second score of a pair) = P(second score of a pair exceeds first score of a pair) If the dependent variable is measured on a continuous scale, this can also be formulated as: H₀: the population median of the difference scores is equal to zero A difference score is the difference between the first score of a pair and the second score of a pair.
Alternative hypothesis	Alternative hypothesis
H₁ two sided: $\pi_1 \neq \pi_2$ H₁ right sided: $\pi_1 > \pi_2$ H₁ left sided: $\pi_1 < \pi_2$	H₁ two sided: P(first score of a pair exceeds second score of a pair) $\neq$ P(second score of a pair exceeds first score of a pair) H₁ right sided: P(first score of a pair exceeds second score of a pair) > P(second score of a pair exceeds first score of a pair) H₁ left sided: P(first score of a pair exceeds second score of a pair) < P(second score of a pair exceeds first score of a pair) If the dependent variable is measured on a continuous scale, this can also be formulated as: H₁ two sided: the population median of the difference scores is different from zero H₁ right sided: the population median of the difference scores is larger than zero H₁ left sided: the population median of the difference scores is smaller than zero
Assumptions	Assumptions
Sample size is large enough for $z$ to be approximately normally distributed. Rule of thumb: Significance test: number of successes and number of failures are each 5 or more in both sample groups Regular (large sample) 90%, 95%, or 99% confidence interval: number of successes and number of failures are each 10 or more in both sample groups Plus four 90%, 95%, or 99% confidence interval: sample sizes of both groups are 5 or more Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Test statistic	Test statistic
$z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$ Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$	$W = $ number of difference scores that is larger than 0
Sampling distribution of $z$ if H₀ were true	Sampling distribution of $W$ if H₀ were true
Approximately the standard normal distribution	The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$. If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1-P)} = \sqrt{n \times 0.5(1 - 0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W - n \times 0.5}{\sqrt{n \times 0.5(1 - 0.5)}}$$ follows approximately the standard normal distribution if the null hypothesis were true.
Significant?	Significant?
Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$	If $n$ is small, the table for the binomial distribution should be used: Two sided: Check if $W$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $W$ observed in sample is in the rejection region or Find right sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $W$ observed in sample is in the rejection region or Find left sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ If $n$ is large, the table for standard normal probabilities can be used: Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$	n.a.
Regular (large sample): $(p_1 - p_2) \pm z^* \times \sqrt{\dfrac{p_1(1 - p_1)}{n_1} + \dfrac{p_2(1 - p_2)}{n_2}}$ where the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval) With plus four method: $(p_{1.plus} - p_{2.plus}) \pm z^* \times \sqrt{\dfrac{p_{1.plus}(1 - p_{1.plus})}{n_1 + 2} + \dfrac{p_{2.plus}(1 - p_{2.plus})}{n_2 + 2}}$ where $p_{1.plus} = \dfrac{X_1 + 1}{n_1 + 2}$, $p_{2.plus} = \dfrac{X_2 + 1}{n_2 + 2}$, and the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval)	-
Equivalent to	Equivalent to
When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels.	Two sided sign test is equivalent to McNemar's test: $W = b$, and $z^2 = X^2$ Friedman test with two related groups
Example context	Example context
Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.	Do people tend to score higher on mental health after a mindfulness course?
SPSS	SPSS
SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to: Analyze > Descriptive Statistics > Crosstabs... Put your independent (grouping) variable in the box below Row(s), and your dependent variable in the box below Column(s) Click the Statistics... button, and click on the square in front of Chi-square Continue and click OK	Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the Sign test
Jamovi	Jamovi
Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to: Frequencies > Independent Samples - $\chi^2$ test of association Put your independent (grouping) variable in the box below Rows, and your dependent variable in the box below Columns	Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to: ANOVA > Repeated Measures ANOVA - Friedman Put the two paired variables in the box below Measures
Practice questions	Practice questions

z test for the difference between two proportions - overview