z test for the difference between two proportions overview

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$z$ test for the difference between two proportions	Pearson correlation	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	Variable 1
One categorical with 2 independent groups	One quantitative of interval or ratio level
Dependent variable	Variable 2
One categorical with 2 independent groups	One quantitative of interval or ratio 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: $\rho = \rho_0$ Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level.
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: $\rho \neq \rho_0$ H₁ right sided: $\rho > \rho_0$ H₁ left sided: $\rho < \rho_0$
Assumptions	Assumptions of test for correlation
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	In the population, the two variables are jointly normally distributed (this covers the normality, homoscedasticity, and linearity assumptions) Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another Note: these assumptions are only important for the significance test and confidence interval, not for the correlation coefficient itself. The correlation coefficient just measures the strength of the linear relationship between two variables.
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 $\pi_2 > \pi_1.$	Test statistic for testing H0: $\rho = 0$: $t = \dfrac{r \times \sqrt{N - 2}}{\sqrt{1 - r^2}} $ where $r$ is the sample correlation $r = \frac{1}{N - 1} \sum_{j}\Big(\frac{x_{j} - \bar{x}}{s_x} \Big) \Big(\frac{y_{j} - \bar{y}}{s_y} \Big)$ and $N$ is the sample size Test statistic for testing values for $\rho$ other than $\rho = 0$: $z = \dfrac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\dfrac{1}{N - 3}}}$ $r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$, where $r$ is the sample correlation $\rho_{0_{Fisher}} = \dfrac{1}{2} \times \log\Bigg( \dfrac{1 + \rho_0}{1 - \rho_0} \Bigg )$, where $\rho_0$ is the population correlation according to H0
Sampling distribution of $z$ if H₀ were true	Sampling distribution of $t$ and of $z$ if H₀ were true
Approximately the standard normal distribution	Sampling distribution of $t$: $t$ distribution with $N - 2$ degrees of freedom Sampling distribution of $z$: Approximately the standard normal distribution
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 $\alpha$	$t$ Test two sided: Check if $t$ observed in sample is at least as extreme as critical value $t^$ or Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ $t$ Test right sided: Check if $t$ observed in sample is equal to or larger than critical value $t^$ or Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ $t$ Test left sided: Check if $t$ observed in sample is equal to or smaller than critical value $t^$ or Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ $z$ Test 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 $\alpha$ $z$ Test 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 $\alpha$ $z$ Test 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 $\pi_1 - \pi_2$	Approximate $C$% confidence interval for $\rho$
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)	First compute the approximate $C$% confidence interval for $\rho_{Fisher}$: $lower_{Fisher} = r_{Fisher} - z^* \times \sqrt{\dfrac{1}{N - 3}}$ $upper_{Fisher} = r_{Fisher} + z^* \times \sqrt{\dfrac{1}{N - 3}}$ where $r_{Fisher} = \frac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$ 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). Then transform back to get the approximate $C$% confidence interval for $\rho$: lower bound = $\dfrac{e^{2 \times lower_{Fisher}} - 1}{e^{2 \times lower_{Fisher}} + 1}$ upper bound = $\dfrac{e^{2 \times upper_{Fisher}} - 1}{e^{2 \times upper_{Fisher}} + 1}$
n.a.	Properties of the Pearson correlation coefficient
-	The Pearson correlation coefficient is a measure for the linear relationship between two quantitative variables. The Pearson correlation coefficient squared reflects the proportion of variance explained in one variable by the other variable. The Pearson correlation coefficient can take on values between -1 (perfect negative relationship) and 1 (perfect positive relationship). A value of 0 means no linear relationship. The absolute size of the Pearson correlation coefficient is not affected by any linear transformation of the variables. However, the sign of the Pearson correlation will flip when the scores on one of the two variables are multiplied by a negative number (reversing the direction of measurement of that variable). For example: the correlation between $x$ and $y$ is equivalent to the correlation between $3x + 5$ and $2y - 6$. the absolute value of the correlation between $x$ and $y$ is equivalent to the absolute value of the correlation between $-3x + 5$ and $2y - 6$. However, the signs of the two correlation coefficients will be in opposite directions, due to the multiplication of $x$ by $-3$. The Pearson correlation coefficient does not say anything about causality. The Pearson correlation coefficient is sensitive to outliers.
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.	OLS regression with one independent variable: $b_1 = r \times \frac{s_y}{s_x}$ Results significance test ($t$ and $p$ value) testing $H_0$: $\beta_1 = 0$ are equivalent to results significance test testing $H_0$: $\rho = 0$
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.	Is there a linear relationship between physical health and mental health?
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 > Correlate > Bivariate... Put your two variables in the box below Variables
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	Regression > Correlation Matrix Put your two variables in the white box at the right Under Correlation Coefficients, select Pearson (selected by default) Under Hypothesis, select your alternative hypothesis
Practice questions	Practice questions

z test for the difference between two proportions - overview