twoProportionInference(k1, n1, k2, n2, CL)
Overview
The twoProportionInference function performs statistical inference on two independent proportions. It calculates the confidence interval and p-value for the difference between two proportions p_1 and p_2 .
Parameters
| Parameter | Type | Description | Default |
|---|---|---|---|
k1 |
Number | Number of successful outcomes in the first sample. | - |
n1 |
Number | Total number of trials in the first sample. | - |
k2 |
Number | Number of successful outcomes in the second sample. | - |
n2 |
Number | Total number of trials in the second sample. | - |
CL |
Number | Confidence level for the confidence interval. | 0.95 |
Returns
| Return | Type | Description |
|---|---|---|
pValue |
Number | The p-value of the Z-test. |
rejectH0 |
Boolean | Whether to reject the null hypothesis at the given alpha. |
stat |
Number | The Z-score of the test. |
pHat |
Table -> Number | Estimated proportions for both samples and overall. |
lowerCI |
Number | Lower bound of the confidence interval for p_1 - p_2 . |
upperCI |
Number | Upper bound of the confidence interval for p_1 - p_2 . |
parametric |
Boolean | Whether the test is parametric (always true for Z-test). |
testType |
String | Specifies the type of test, "Two Proportion Test". |
statType |
String | Specifies the type of statistic used, "Z". |
warning |
Boolean | Whether the sample size is too small for a reliable test. |
Example
local k1 = 50
local n1 = 100
local k2 = 40
local n2 = 90
local CL = 0.95
local result = twoProportionInference(k1, n1, k2, n2, CL)
print(result.pValue, result.rejectH0, result.stat, result.lowerCI, result.upperCI) -- Output will vary based on the input
Mathematical Background
The pooled proportion \hat{p} is calculated as:
\hat{p} = \frac{k_1 + k_2}{n_1 + n_2}
The standard error SE is calculated using:
SE = \sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}
The confidence interval is given by:
\text{Lower CI} = \text{max}\left(\hat{p}_1 - \hat{p}_2 - Z_{\alpha/2} \times \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}, -1\right)
\text{Upper CI} = \text{min}\left(\hat{p}_1 - \hat{p}_2 + Z_{\alpha/2} \times \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}, 1\right)
The Z-score Z and p-value are also calculated based on the above statistics.