Welcome to your essential AP Statistics digital toolkit! This page is designed to act as a "cheat sheet" to help you perform and verify common statistical calculations quickly. Use these calculators to check your homework, understand the mechanics of formulas, and build confidence before the AP exam. Remember, understanding the conditions and interpretations is just as important as the calculation itself!
1-Sample Z-Test for a Proportion
This test is used to determine if there is significant evidence to reject a claim about a population proportion (p₀) based on a single sample. Before using this test, always check the conditions: Random, 10% Condition, and Large Counts (np₀ ≥ 10 and n(1-p₀) ≥ 10).
The formula for the z-statistic is: z = (p̂ - p₀) / √((p₀ * (1 - p₀)) / n)
Z-Test Calculator
Example
A national poll claims that 60% of citizens support a new policy. A local politician surveys 100 constituents and finds that 64 of them support the policy. Is there significant evidence to suggest the support in this district is different from the national claim?
p̂: 64/100 = 0.64
p₀: 0.60
n: 100
Enter these values into the calculator to find the z-statistic and p-value.
1-Sample T-Test for a Mean
Use this test when you want to check a claim about a population mean (μ₀) using a sample mean (x̄), but you do not know the population standard deviation (σ). Instead, you use the sample standard deviation (s). Conditions: Random, 10% Condition, and Normal/Large Sample (n ≥ 30 or population is stated to be normal).
The formula for the t-statistic is: t = (x̄ - μ₀) / (s / √n)
T-Test Calculator
Example
A coffee shop claims its large coffees contain an average of 15 oz. A customer suspects they are being underfilled. They measure 40 large coffees and find a sample mean of 14.6 oz with a sample standard deviation of 0.8 oz. Is there significant evidence the mean volume is less than 15 oz?
x̄: 14.6
μ₀: 15
s: 0.8
n: 40
Confidence Interval for a Proportion
A confidence interval provides a range of plausible values for an unknown population proportion. The formula is: p̂ ± z* * √((p̂ * (1 - p̂)) / n), where z* is the critical value for your chosen confidence level.
CI for Proportion Calculator
90%
95%
99%
Example
In a random sample of 500 high school students, 125 said they have a part-time job. Construct a 95% confidence interval for the proportion of all high school students who have a part-time job.
p̂: 125/500 = 0.25
n: 500
Confidence Level: 95%
// Helper function for Normal CDF approximation (for Z-test p-value)
function normalCdf(x) {
// Abramowitz and Stegun approximation for the error function erf(x)
var t = 1 / (1 + 0.3275911 * Math.abs(x));
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var erf = 1 – (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-x * x);
if (x < 0) {
erf = -erf;
}
// CDF of standard normal is 0.5 * (1 + erf(x / sqrt(2)))
return 0.5 * (1 + erf(x / Math.sqrt(2)));
}
function calculateZTestProp() {
var pHat = parseFloat(document.getElementById('z_pHat').value);
var p0 = parseFloat(document.getElementById('z_p0').value);
var n = parseInt(document.getElementById('z_n').value);
var resultDiv = document.getElementById('zTestResult');
if (isNaN(pHat) || isNaN(p0) || isNaN(n) || n <= 0 || pHat 1 || p0 1) {
resultDiv.innerHTML = 'Please enter valid numbers. Proportions must be between 0 and 1, and sample size must be positive.';
resultDiv.style.display = 'block';
return;
}
var standardError = Math.sqrt((p0 * (1 – p0)) / n);
if (standardError === 0) {
resultDiv.innerHTML = 'Calculation failed. The hypothesized proportion cannot be 0 or 1.';
resultDiv.style.display = 'block';
return;
}
var z_statistic = (pHat – p0) / standardError;
// P-value calculation
var p_value_one_sided_less = normalCdf(z_statistic);
var p_value_one_sided_greater = 1 – p_value_one_sided_less;
var p_value_two_sided = 2 * (z_statistic > 0 ? p_value_one_sided_greater : p_value_one_sided_less);
resultDiv.innerHTML = '
Calculation Results:
' +
'Standard Error: ' + standardError.toFixed(5) + '' +
'Z-Statistic: ' + z_statistic.toFixed(4) + '' +
'P-Value (one-sided, <): ' + p_value_one_sided_less.toFixed(5) + '' +
'P-Value (one-sided, >): ' + p_value_one_sided_greater.toFixed(5) + '' +
'P-Value (two-sided, ≠): ' + p_value_two_sided.toFixed(5) + '';
resultDiv.style.display = 'block';
}
function calculateTTestMean() {
var xBar = parseFloat(document.getElementById('t_xBar').value);
var mu0 = parseFloat(document.getElementById('t_mu0').value);
var s = parseFloat(document.getElementById('t_s').value);
var n = parseInt(document.getElementById('t_n').value);
var resultDiv = document.getElementById('tTestResult');
if (isNaN(xBar) || isNaN(mu0) || isNaN(s) || isNaN(n) || n <= 1 || s < 0) {
resultDiv.innerHTML = 'Please enter valid numbers. Sample size (n) must be greater than 1 and standard deviation (s) cannot be negative.';
resultDiv.style.display = 'block';
return;
}
var standardError = s / Math.sqrt(n);
if (standardError === 0) {
resultDiv.innerHTML = 'Calculation failed. Sample standard deviation cannot be zero for this test.';
resultDiv.style.display = 'block';
return;
}
var t_statistic = (xBar – mu0) / standardError;
var df = n – 1;
resultDiv.innerHTML = '
Calculation Results:
' +
'Degrees of Freedom (df): ' + df + '' +
'Standard Error (s/√n): ' + standardError.toFixed(5) + '' +
'T-Statistic: ' + t_statistic.toFixed(4) + '' +
'To find the p-value, use a t-distribution table or a calculator function (like tcdf) with ' + df + ' degrees of freedom.';
resultDiv.style.display = 'block';
}
function calculateCIProp() {
var pHat = parseFloat(document.getElementById('ci_pHat').value);
var n = parseInt(document.getElementById('ci_prop_n').value);
var confLevel = document.getElementById('ci_prop_conf').value;
var resultDiv = document.getElementById('ciPropResult');
if (isNaN(pHat) || isNaN(n) || n <= 0 || pHat 1) {
resultDiv.innerHTML = 'Please enter valid numbers. Proportion must be between 0 and 1, and sample size must be positive.';
resultDiv.style.display = 'block';
return;
}
var z_star;
switch (confLevel) {
case '90':
z_star = 1.645;
break;
case '95':
z_star = 1.96;
break;
case '99':
z_star = 2.576;
break;
default:
z_star = 1.96; // Default to 95%
}
var standardError = Math.sqrt((pHat * (1 – pHat)) / n);
var marginOfError = z_star * standardError;
var lowerBound = pHat – marginOfError;
var upperBound = pHat + marginOfError;
resultDiv.innerHTML = '
' + confLevel + '% Confidence Interval Results:
' +
'Critical Value (z*): ' + z_star + '' +
'Standard Error: ' + standardError.toFixed(5) + '' +
'Margin of Error: ' + marginOfError.toFixed(5) + '' +
'Confidence Interval: (' + lowerBound.toFixed(4) + ', ' + upperBound.toFixed(4) + ')' +
'We are ' + confLevel + '% confident that the true population proportion is between ' + (lowerBound*100).toFixed(2) + '% and ' + (upperBound*100).toFixed(2) + '%.';
resultDiv.style.display = 'block';
}