Test Statistic Calculator
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Understanding the Test Statistic
In inferential statistics, a test statistic is a value calculated from sample data that helps determine whether to reject or fail to reject a null hypothesis. It measures how far the sample result deviates from the null hypothesis, assuming the null hypothesis is true. The calculation of the test statistic depends on the type of data, the sample size, and the hypothesis being tested.
Common Types of Test Statistics
1. Z-Test
The Z-test is used when the population standard deviation is known or when the sample size is large (typically n > 30). It assumes that the data are normally distributed.
Formula for Z-test for a Mean:
Z = (x̄ - μ₀) / (σ / √n)
Where:
x̄ is the sample mean.μ₀ is the hypothesized population mean.σ is the population standard deviation.n is the sample size.
Formula for Z-test for a Proportion:
Z = (p̂ - p₀) / √(p₀(1-p₀)/n)
Where:
p̂ is the sample proportion.p₀ is the hypothesized population proportion.n is the sample size.
(This calculator currently focuses on the Z-test for a mean.)
2. T-Test
The T-test is used when the population standard deviation is unknown and the sample size is small (typically n < 30). It also assumes data are approximately normally distributed. The t-distribution is used instead of the normal distribution.
Formula for T-test for a Mean:
t = (x̄ - μ₀) / (s / √n)
Where:
x̄ is the sample mean.μ₀ is the hypothesized population mean.s is the sample standard deviation.n is the sample size.
The degrees of freedom for a one-sample t-test is
df = n - 1.
3. Chi-Square (χ²) Test
The Chi-Square test is used for categorical data. It assesses the relationship between observed frequencies and expected frequencies.
Formula for Chi-Square Test Statistic:
χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ]
Where:
Oᵢ are the observed frequencies for each category.Eᵢ are the expected frequencies for each category.Σ denotes the sum across all categories.
The degrees of freedom for a Chi-Square test depend on the specific application (e.g., for goodness-of-fit, df = number of categories – 1; for independence, df = (number of rows – 1) * (number of columns – 1)).
How to Use This Calculator
- Select the Type of Test Statistic you wish to calculate.
- Input the relevant values for the chosen test type (Sample Mean, Hypothesized Mean, Standard Deviations, Sample Size, Observed/Expected Frequencies). Ensure you enter numeric values. For Chi-Square, enter frequencies separated by commas.
- Click the "Calculate Test Statistic" button.
- The calculated test statistic will be displayed.
Use Cases
- Z-Test: Comparing a sample mean to a known population mean when population variance is known, or determining if a sample proportion differs significantly from a hypothesized population proportion (e.g., checking if a new drug's effectiveness differs from a known baseline).
- T-Test: Comparing a sample mean to a known population mean when population variance is unknown and the sample is small (e.g., evaluating the average score of a small group of students on a new teaching method).
- Chi-Square Test: Determining if there is a significant association between two categorical variables (e.g., is there a relationship between gender and preferred political party?) or if observed categorical data fits an expected distribution (e.g., do observed dice roll frequencies match the expected uniform distribution?).
function showInputs(type) {
document.getElementById('zTestInputs').style.display = (type === 'z-test') ? 'block' : 'none';
document.getElementById('tTestInputs').style.display = (type === 't-test') ? 'block' : 'none';
document.getElementById('chiSquareInputs').style.display = (type === 'chi-square') ? 'block' : 'none';
}
function calculateTestStatistic() {
var statisticType = document.getElementById('statisticType').value;
var result = '–';
var unit = 'N/A';
if (statisticType === 'z-test') {
var sampleMean = parseFloat(document.getElementById('sampleMean').value);
var populationMean = parseFloat(document.getElementById('populationMean').value);
var populationStdDev = parseFloat(document.getElementById('populationStdDev').value);
var sampleSize = parseFloat(document.getElementById('sampleSize').value);
if (isNaN(sampleMean) || isNaN(populationMean) || isNaN(populationStdDev) || isNaN(sampleSize) || populationStdDev <= 0 || sampleSize <= 0) {
result = "Invalid input";
} else {
var z = (sampleMean – populationMean) / (populationStdDev / Math.sqrt(sampleSize));
result = z.toFixed(4);
unit = 'Z-score';
}
} else if (statisticType === 't-test') {
var tSampleMean = parseFloat(document.getElementById('tSampleMean').value);
var tHypothesizedMean = parseFloat(document.getElementById('tHypothesizedMean').value);
var sampleStdDev = parseFloat(document.getElementById('sampleStdDev').value);
var tSampleSize = parseFloat(document.getElementById('tSampleSize').value);
if (isNaN(tSampleMean) || isNaN(tHypothesizedMean) || isNaN(sampleStdDev) || isNaN(tSampleSize) || sampleStdDev <= 0 || tSampleSize <= 1) {
result = "Invalid input";
} else {
var t = (tSampleMean – tHypothesizedMean) / (sampleStdDev / Math.sqrt(tSampleSize));
result = t.toFixed(4);
unit = 'T-score';
}
} else if (statisticType === 'chi-square') {
var observedFrequenciesStr = document.getElementById('observedFrequencies').value;
var expectedFrequenciesStr = document.getElementById('expectedFrequencies').value;
var observedFrequencies = observedFrequenciesStr.split(',').map(function(item) { return parseFloat(item.trim()); });
var expectedFrequencies = expectedFrequenciesStr.split(',').map(function(item) { return parseFloat(item.trim()); });
if (observedFrequencies.some(isNaN) || expectedFrequencies.some(isNaN) || observedFrequencies.length === 0 || observedFrequencies.length !== expectedFrequencies.length) {
result = "Invalid input";
} else {
var chiSquareStat = 0;
var allExpectedPositive = true;
for (var i = 0; i < observedFrequencies.length; i++) {
if (expectedFrequencies[i] 0)";
} else {
result = chiSquareStat.toFixed(4);
unit = 'Chi-Square (χ²) value';
}
}
}
document.getElementById('testStatisticResult').innerText = result;
document.getElementById('resultUnit').innerText = 'Unit: ' + unit;
}
// Initial setup and event listener
document.addEventListener('DOMContentLoaded', function() {
showInputs(document.getElementById('statisticType').value);
document.getElementById('statisticType').addEventListener('change', function() {
showInputs(this.value);
});
});