z test for a single proportion - overview

• H₀: the population medians for the $I$ groups are equal

• H₀: the population scores in any of the $I$ groups are not systematically higher or lower than the population scores in any of the other groups

• H₀: P(an observation from population $g$ exceeds an observation from population $h$) = P(an observation from population $h$ exceeds an observation from population $g$), for each pair of groups.

• H₀: the distribution of the dependent variable is the same in each of the $I$ populations

• H₀: the row and column variables are independent

• H₁: not all of the population medians for the $I$ groups are equal

• H₁: the poplation scores in some groups are systematically higher or lower than the population scores in other groups

• H₁: for at least one pair of groups:
  P(an observation from population $g$ exceeds an observation from population $h$) $\neq$ P(an observation from population $h$ exceeds an observation from population $g$)

• H₁: the distribution of the dependent variable is not the same in all of the $I$ populations

• H₁: the row and column variables are dependent

• 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

• Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, 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

$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$

Here $N$ is the total sample size, $R_i$ is the sum of ranks in group $i$, and $n_i$ is the sample size of group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N (N + 1)} \times \sum \frac{R^2_i}{n_i}$ and then subtract $3(N + 1)$.

For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom.

For small samples, the exact distribution of $H$ should be used.

• Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
• Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

• Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
• Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

• Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
• Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

• 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 $\alpha$

• 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 $\alpha$

• $p \pm z^* \times \sqrt{\dfrac{p(1 - p)}{N}}$
  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)

• $p_{plus} \pm z^* \times \sqrt{\dfrac{p_{plus}(1 - p_{plus})}{N + 4}}$
  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)

• 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.

• 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

• Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
• Click on the Define Range... button. If you can't click on it, first click on the grouping variable so its background turns yellow
• Fill in the smallest value you have used to indicate your groups in the box next to Minimum, and the largest value you have used to indicate your groups in the box next to Maximum
• Continue and click OK

• 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

• 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

• Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable

• Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns