One sample z test for the mean - overview

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One sample $z$ test for the mean
One sample Wilcoxon signed-rank test
Binomial test for a single proportion$z$ test for a single proportion$z$ test for the difference between two proportionsGoodness of fit testChi-squared test for the relationship between two categorical variablesOne sample $z$ test for the meanOne sample $t$ test for the meanPaired sample $t$ testTwo sample $z$ testTwo sample $t$ test - equal variances not assumedTwo sample $t$ test - equal variances assumedOne way ANOVATwo way ANOVAPearson correlationRegression (OLS)Logistic regressionMann-Whitney-Wilcoxon testKruskal-Wallis testSign testMcNemar's testCochran's Q testMarginal Homogeneity test / Stuart-Maxwell test Friedman testWilcoxon signed-rank testOne sample Wilcoxon signed-rank testSpearman's rho
Independent variableIndependent variable
NoneNone
Dependent variableDependent variable
One quantitative of interval or ratio levelOne of ordinal level
Null hypothesisNull hypothesis
H0: $\mu = \mu_0$

$\mu$ is the population mean; $\mu_0$ is the population mean according to the null hypothesis		H0: $m = m_0$

$m$ is the population median; $m_0$ is the population median according to the null hypothesis.
Alternative hypothesisAlternative hypothesis
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$ 		H1 two sided: $m \neq m_0$
H1 right sided: $m > m_0$
H1 left sided: $m < m_0$
AssumptionsAssumptions
  • 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
  • The population distribution of the scores is symmetric
  • Sample is a simple random sample from the population. That is, observations are independent of one another
Test statisticTest statistic
$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, $N$ 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 $\mu_0$.		Two different types of test statistics can be used; both will result in the same test outcome. We will denote the first option the $W_1$ statistic (also known as the $T$ statistic), and the second option the $W_2$ statistic. In order to compute each of the test statistics, follow the steps below:
  1. For each subject, compute the sign of the difference score $\mbox{sign}_d = \mbox{sgn}(\mbox{score} - m_0)$. The sign is 1 if the difference is larger than zero, -1 if the diffence is smaller than zero, and 0 if the difference is equal to zero.
  2. For each subject, compute the absolute value of the difference score $|\mbox{score} - m_0|$.
  3. Exclude subjects with a difference score of zero. This leaves us with a remaining number of difference scores equal to $N_r$.
  4. Assign ranks $R_d$ to the $N_r$ remaining absolute difference scores. The smallest absolute difference score corresponds to a rank score of 1, and the largest absolute difference score corresponds to a rank score of $N_r$. If there are ties, assign them the average of the ranks they occupy.
Then compute the test statistic:

  • $W_1 = \sum\, R_d^{+}$
    or
    $W_1 = \sum\, R_d^{-}$
    That is, sum all ranks corresponding to a positive difference or sum all ranks corresponding to a negative difference. Theoratically, both definitions will result in the same test outcome. However:
    • Tables with critical values for $W_1$ are usually based on the smaller of $\sum\, R_d^{+}$ and $\sum\, R_d^{-}$. So if you are using such a table, pick the smaller one.
    • If you are using the normal approximation to find the $p$ value, it makes things most straightforward if you use $W_1 = \sum\, R_d^{+}$ (if you use $W_1 = \sum\, R_d^{-}$, the right and left sided alternative hypotheses 'flip').
  • $W_2 = \sum\, \mbox{sign}_d \times R_d$
    That is, for each remaining difference score, multiply the rank of the absolute difference score by the sign of the difference score, and then sum all of the products.
Sampling distribution of $z$ if H0 were trueSampling distribution of $W_1$ and of $W_2$ if H0 were true
Standard normal distributionSampling distribution of $W_1$:
If $N_r$ is large, $W_1$ is approximately normally distributed with mean $\mu_{W_1}$ and standard deviation $\sigma_{W_1}$ if the null hypothesis were true. Here $$\mu_{W_1} = \frac{N_r(N_r + 1)}{4}$$ $$\sigma_{W_1} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_1 - \mu_{W_1}}{\sigma_{W_1}}$$ follows approximately the standard normal distribution if the null hypothesis were true.

Sampling distribution of $W_2$:
If $N_r$ is large, $W_2$ is approximately normally distributed with mean $0$ and standard deviation $\sigma_{W_2}$ if the null hypothesis were true. Here $$\sigma_{W_2} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_2}{\sigma_{W_2}}$$ follows approximately the standard normal distribution if the null hypothesis were true.

If $N_r$ is small, the exact distribution of $W_1$ or $W_2$ should be used.

Note: the formula for the standard deviations $\sigma_{W_1}$ and $\sigma_{W_2}$ is more complicated if ties are present in the data.
Significant?Significant?
Two sided: Right sided: Left sided: For large samples, the table for standard normal probabilities can be used:
Two sided: Right sided: Left sided:
$C\%$ confidence interval for $\mu$n.a.
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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. 		-
Effect sizen.a.
Cohen's $d$:
Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0$		-
Visual representationn.a.
One sample z test
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Example contextExample context
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.Is the median mental health score different from 50?
n.a.SPSS
-Specify the measurement level of your variable on the Variable View tab, in the column named Measure. Then go to:

Analyze > Nonparametric Tests > One Sample...
  • On the Objective tab, choose Customize Analysis
  • On the Fields tab, specify the variable for which you want to compute the Wilcoxon signed-rank test
  • On the Settings tab, choose Customize tests and check the box for 'Compare median to hypothesized (Wilcoxon signed-rank test)'. Fill in your $m_0$ in the box next to Hypothesized median
  • Click Run
  • Double click on the output table to see the full results
n.a.Jamovi
-T-Tests > One Sample T-Test
  • Put your variable in the box below Dependent Variables
  • Under Tests, select Wilcoxon rank
  • Under Hypothesis, fill in the value for $m_0$ in the box next to Test Value, and select your alternative hypothesis
Practice questionsPractice questions