One sample z test for the mean - overview

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One sample $z$ test for the mean
Chi-squared test for the relationship between two categorical variables
Independent variableIndependent /column variable
NoneOne categorical with $I$ independent groups ($I \geqslant 2$)
Dependent variableDependent /row variable
One quantitative of interval or ratio levelOne categorical with $J$ independent groups ($J \geqslant 2$)
Null hypothesisNull hypothesis
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis
  • There is no association between the row and column variable
    More precise statement:
    • If there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
      The distribution of the dependent variable is the same in each of the $I$ populations
    • If there is one random sample of size $N$ from the total population:
      The row and column variables are independent
Alternative hypothesisAlternative hypothesis
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
  • There is an association between the row and column variable
    More precise statement:
    • If there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
      The distribution of the dependent variable is not the same in all of the $I$ populations
    • If there is one random sample of size $N$ from the total population:
      The row and column variables are dependent
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
  • Sample size is large enough for $X^2$ to be approximately chi-squared distributed under the null hypothesis. Rule of thumb:
    • 2 $\times$ 2 table: all four expected cell counts are 5 or more
    • Larger than 2 $\times$ 2 tables: average of the expected cell counts is 5 or more, smallest expected cell count is 1 or more
  • There are $I$ independent simple random samples from each of $I$ populations defined by the independent variable, or there is one simple random sample from the total population
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 H0, $\sigma$ 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 $\mu_0$
$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells
Sampling distribution of $z$ if H0 were trueSampling distribution of $X^2$ if H0 were true
Standard normalApproximately a chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom
Significant?Significant?
Two sided: Right sided: Left sided:
  • Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or
  • Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
$C\%$ confidence interval for $\mu$n.a.
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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.
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Effect sizen.a.
Cohen's $d$:
Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ 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 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 there an association between economic class and gender? Is the distribution of economic class different between men and women?
n.a.SPSS
-Analyze > Descriptive Statistics > Crosstabs...
  • Put one of your two categorical variables in the box below Row(s), and the other categorical 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
n.a.Jamovi
-Frequencies > Independent Samples - $\chi^2$ test of association
  • Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns
Practice questionsPractice questions