# One sample z test for the mean - overview

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One sample $z$ test for the mean
Two sample $t$ test - equal variances assumed
Independent variableIndependent/grouping variable
NoneOne categorical with 2 independent groups
Dependent variableDependent variable
One quantitative of interval or ratio levelOne quantitative of interval or ratio 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: $\mu_1 = \mu_2$

$\mu_1$ is the population mean for group 1, $\mu_2$ is the population mean for group 2
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: $\mu_1 \neq \mu_2$
H1 right sided: $\mu_1 > \mu_2$
H1 left sided: $\mu_1 < \mu_2$
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
• Within each population, the scores on the dependent variable are normally distributed
• The standard deviation of the scores on the dependent variable is the same in both populations: $\sigma_1 = \sigma_2$
• 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
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 $\mu_0$.
$t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
$\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis.

The denominator $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ is removed from 0.

Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.
n.a.Pooled standard deviation
-$s_p = \sqrt{\dfrac{(n_1 - 1) \times s^2_1 + (n_2 - 1) \times s^2_2}{n_1 + n_2 - 2}}$
Sampling distribution of $z$ if H0 were trueSampling distribution of $t$ if H0 were true
Standard normal distribution$t$ distribution with $n_1 + n_2 - 2$ degrees of freedom
Significant?Significant?
Two sided:
Right sided:
Left sided:
Two sided:
Right sided:
Left sided:
$C\%$ confidence interval for $\mu$$C\% confidence interval for \mu_1 - \mu_2 \bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}} where z^* is the value under the normal curve with the area C / 100 between -z^* and z^* (e.g. z^* = 1.96 for a 95% confidence interval) The confidence interval for \mu can also be used as significance test. (\bar{y}_1 - \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}} where the critical value t^* is the value under the t_{n_1 + n_2 - 2} distribution with the area C / 100 between -t^* and t^* (e.g. t^* = 2.086 for a 95% confidence interval when df = 20) The confidence interval for \mu_1 - \mu_2 can also be used as significance test. Effect sizeEffect size Cohen's d: Standardized difference between the sample mean and \mu_0:$$d = \frac{\bar{y} - \mu_0}{\sigma}$$Indicates how many standard deviations \sigma the sample mean \bar{y} is removed from \mu_0 Cohen's d: Standardized difference between the mean in group 1 and in group 2:$$d = \frac{\bar{y}_1 - \bar{y}_2}{s_p}$$Indicates how many standard deviations$s_p$the two sample means are removed from each other Visual representationVisual representation n.a.Equivalent to -One way ANOVA with an independent variable with 2 levels ($I$= 2): • two sided two sample$t$test equivalent to ANOVA$F$test when$I$= 2 • two sample$t$test equivalent to$t$test for contrast when$I$= 2 • two sample$t$test equivalent to$t$test multiple comparisons when$I$= 2 OLS regression with one categorical independent variable with 2 levels: • two sided two sample$t$test equivalent to$F$test regression model • two sample$t$test equivalent to$t$test for regression coefficient$\beta_1$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 average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women.
n.a.SPSS
-Analyze > Compare Means > Independent-Samples T Test...
• Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable
• Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
• Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
• Continue and click OK
n.a.Jamovi
-T-Tests > Independent Samples T-Test
• Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
• Under Tests, select Student's (selected by default)
• Under Hypothesis, select your alternative hypothesis
Practice questionsPractice questions