One sample z test for the mean - overview

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One sample $z$ test for the mean
ANCOVA
Independent variableIndependent variables
NoneOne or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)
Dependent variableDependent variable
One quantitative of interval or ratio levelOne quantitative of interval or ratio level
Null hypothesisTHIS TABLE IS YET TO BE COMPLETED
H0: $\mu = \mu_0$

Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.
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Alternative hypothesisn.a.
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
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Assumptionsn.a.
  • 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
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Test statisticn.a.
$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
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 $\mu_0$.
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Sampling distribution of $z$ if H0 were truen.a.
Standard normal distribution-
Significant?n.a.
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.
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Effect sizen.a.
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 $\mu_0.$
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Visual representationn.a.
One sample z test
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Example contextn.a.
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 $\sigma = 3.$-
Practice questionsPractice questions