One Sample t-Test Calculator

Understanding the One-Sample T-Test

The one-sample t-test is used to determine whether the mean of a sample significantly differs from a hypothesized population mean. This test is useful for comparing the average result of a sample to a known value when the population variance is unknown.

Key Components of the One-Sample T-Test

• Sample Mean (\( \bar{X} \)): The average value of the sample data.
• Hypothesized Mean (\( \mu_0 \)): The population mean we are testing against.
• t-Score: A value derived from the sample data that allows us to compare the sample mean to the hypothesized mean.

Formula for the One-Sample T-Test

The t-score for a one-sample t-test is calculated as:

where:

• \( \bar{X} \): Sample mean
• \( \mu_0 \): Hypothesized population mean
• \( s \): Standard deviation of the sample
• \( n \): Sample size

Programmatically Calculating the One-Sample T-Test

Below are examples for calculating the one-sample t-test in JavaScript, Python, and R, including left-tailed, right-tailed, and two-tailed versions.

1. Using JavaScript (with jStat)

In JavaScript, the jStat library can be used to calculate the one-sample t-test:

// Define inputs
const sampleData = [20, 22, 24, 21, 19, 25];
const hypMean = 23;  // Hypothesized mean
const n = sampleData.length;
const sampleMean = jStat.mean(sampleData);
const sampleStdDev = jStat.stdev(sampleData, true);

// Calculate t-score
const tScore = (sampleMean - hypMean) / (sampleStdDev / Math.sqrt(n));
const df = n - 1;
const alpha = 0.05;  // Significance level

// Calculate p-values
const pValueTwoTailed = 2 * (1 - jStat.studentt.cdf(Math.abs(tScore), df));
const pValueRightTailed = 1 - jStat.studentt.cdf(tScore, df);
const pValueLeftTailed = jStat.studentt.cdf(tScore, df);

console.log(`Two-tailed p-value: ${pValueTwoTailed.toFixed(5)}`);
console.log(`Right-tailed p-value: ${pValueRightTailed.toFixed(5)}`);
console.log(`Left-tailed p-value: ${pValueLeftTailed.toFixed(5)}`);

2. Using Python (with SciPy)

In Python, the SciPy library can be used to calculate the one-sample t-test:

import numpy as np
from scipy.stats import t

# Define inputs
sample_data = np.array([20, 22, 24, 21, 19, 25])
hyp_mean = 23  # Hypothesized mean
n = len(sample_data)
sample_mean = np.mean(sample_data)
sample_stddev = np.std(sample_data, ddof=1)

# Calculate t-score
t_score = (sample_mean - hyp_mean) / (sample_stddev / np.sqrt(n))
df = n - 1
alpha = 0.05  # Significance level

# Calculate p-values
p_value_two_tailed = 2 * (1 - t.cdf(abs(t_score), df))
p_value_right_tailed = 1 - t.cdf(t_score, df)
p_value_left_tailed = t.cdf(t_score, df)

print(f"Two-tailed p-value: {p_value_two_tailed:.5f}")
print(f"Right-tailed p-value: {p_value_right_tailed:.5f}")
print(f"Left-tailed p-value: {p_value_left_tailed:.5f}")

3. Using R

In R, the stats package includes functions to calculate the one-sample t-test:

# Define inputs
sample_data <- c(20, 22, 24, 21, 19, 25)
hyp_mean <- 23  # Hypothesized mean
n <- length(sample_data)
sample_mean <- mean(sample_data)
sample_stddev <- sd(sample_data)

# Calculate t-score
t_score <- (sample_mean - hyp_mean) / (sample_stddev / sqrt(n))
df <- n - 1
alpha <- 0.05  # Significance level

# Calculate p-values
p_value_two_tailed <- 2 * (1 - pt(abs(t_score), df))
p_value_right_tailed <- 1 - pt(t_score, df)
p_value_left_tailed <- pt(t_score, df)

cat("Two-tailed p-value:", round(p_value_two_tailed, 5), "\n")
cat("Right-tailed p-value:", round(p_value_right_tailed, 5), "\n")
cat("Left-tailed p-value:", round(p_value_left_tailed, 5), "\n")

Further Reading

• Student's T-Test on Wikipedia
• Explore More Calculators on the Research Scientist Pod