# z test for a single proportion - overview

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$z$ test for a single proportion
One sample $z$ test for the mean
Independent variableIndependent variable
NoneNone
Dependent variableDependent variable
One categorical with 2 independent groupsOne quantitative of interval or ratio level
Null hypothesisNull hypothesis
H0: $\pi = \pi_0$

Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.
H0: $\mu = \mu_0$

Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.
Alternative hypothesisAlternative hypothesis
H1 two sided: $\pi \neq \pi_0$
H1 right sided: $\pi > \pi_0$
H1 left sided: $\pi < \pi_0$
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
AssumptionsAssumptions
• Sample size is large enough for $z$ to be approximately normally distributed. Rule of thumb:
• Significance test: $N \times \pi_0$ and $N \times (1 - \pi_0)$ are each larger than 10
• Regular (large sample) 90%, 95%, or 99% confidence interval: number of successes and number of failures in sample are each 15 or more
• Plus four 90%, 95%, or 99% confidence interval: total sample size is 10 or more
• Sample is a simple random sample from the population. That is, observations are independent of one another
If the sample size is too small for $z$ to be approximately normally distributed, the binomial test for a single proportion should be used.
• Scores are normally distributed in the population
• Population standard deviation $\sigma$ is known
• Sample is a simple random sample from the population. That is, observations are independent of one another
Test statisticTest statistic
$z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}$
Here $p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size, and $\pi_0$ is the population proportion of successes according to the null hypothesis.
$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size.

The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$.
Sampling distribution of $z$ if H0 were trueSampling distribution of $z$ if H0 were true
Approximately the standard normal distributionStandard normal distribution
Significant?Significant?
Two sided:
Right sided:
Left sided:
Two sided:
Right sided:
Left sided:
Approximate $C\%$ confidence interval for $\pi$$C\% confidence interval for \mu Regular (large sample): • p \pm z^* \times \sqrt{\dfrac{p(1 - p)}{N}} where the critical value z^* is the value under the normal curve with the area C / 100 between -z^* and z^* (e.g. z^* = 1.96 for a 95% confidence interval) With plus four method: • p_{plus} \pm z^* \times \sqrt{\dfrac{p_{plus}(1 - p_{plus})}{N + 4}} where p_{plus} = \dfrac{X + 2}{N + 4} and the critical value z^* is the value under the normal curve with the area C / 100 between -z^* and z^* (e.g. z^* = 1.96 for a 95% confidence interval) \bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}} where the critical value z^* is the value under the normal curve with the area C / 100 between -z^* and z^* (e.g. z^* = 1.96 for a 95% confidence interval). The confidence interval for \mu can also be used as significance test. n.a.Effect size -Cohen's d: Standardized difference between the sample mean and \mu_0:$$d = \frac{\bar{y} - \mu_0}{\sigma}$$Cohen's$d$indicates how many standard deviations$\sigma$the sample mean$\bar{y}$is removed from$\mu_0.$n.a.Visual representation - Equivalent ton.a. • When testing two sided: goodness of fit test, with a categorical variable with 2 levels. • When$N$is large, the$p$value from the$z$test for a single proportion approaches the$p$value from the binomial test for a single proportion. The$z$test for a single proportion is just a large sample approximation of the binomial test for a single proportion. - Example contextExample context Is the proportion of smokers amongst office workers different from$\pi_0 = 0.2$? Use the normal approximation for the sampling distribution of the test statistic.Is the average mental health score of office workers different from$\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is$\sigma = 3.$SPSSn.a. Analyze > Nonparametric Tests > Legacy Dialogs > Binomial... • Put your dichotomous variable in the box below Test Variable List • Fill in the value for$\pi_0$in the box next to Test Proportion If computation time allows, SPSS will give you the exact$p$value based on the binomial distribution, rather than the approximate$p$value based on the normal distribution - Jamovin.a. Frequencies > 2 Outcomes - Binomial test • Put your dichotomous variable in the white box at the right • Fill in the value for$\pi_0$in the box next to Test value • Under Hypothesis, select your alternative hypothesis Jamovi will give you the exact$p$value based on the binomial distribution, rather than the approximate$p\$ value based on the normal distribution
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Practice questionsPractice questions