Student's t-Calculator
Use this calculator to determine the t-statistic and degrees of freedom for a one-sample t-test. This test is used to compare a sample mean to a known or hypothesized population mean when the population standard deviation is unknown.
Understanding the Student's t-Test
The Student's t-test is a fundamental inferential statistical test used to determine if there is a significant difference between the means of two groups, or between a sample mean and a hypothesized population mean. This calculator focuses on the one-sample t-test, which is particularly useful when you want to assess whether a sample mean is significantly different from a known or theoretical population mean, and the population standard deviation is unknown.
When to Use a One-Sample t-Test
You would typically use a one-sample t-test in situations like:
- Comparing the average test score of a class to a national average.
- Evaluating if the mean weight of a product batch deviates from a specified target weight.
- Assessing if the average reaction time of a group differs from a known baseline.
A key assumption for the t-test is that the data is approximately normally distributed, especially for smaller sample sizes (n < 30). For larger sample sizes, the Central Limit Theorem helps ensure the sampling distribution of the mean is approximately normal, even if the original data is not.
Inputs Explained
- Sample Mean (x̄): This is the average value of your collected data points from the sample. It's calculated by summing all values in your sample and dividing by the number of values.
- Hypothesized Population Mean (μ): This is the value you are comparing your sample mean against. It could be a known population average, a target value, or a theoretical expectation.
- Sample Standard Deviation (s): This measures the amount of variation or dispersion of your data points around the sample mean. A larger standard deviation indicates more spread-out data.
- Sample Size (n): This is the total number of observations or data points in your sample.
The t-Statistic Formula
The t-statistic is calculated using the following formula:
t = (x̄ - μ) / (s / √n)
Where:
x̄= Sample Meanμ= Hypothesized Population Means= Sample Standard Deviationn= Sample Size√n= Square root of the Sample Size
The term s / √n is known as the Standard Error of the Mean, which estimates the standard deviation of the sampling distribution of the sample mean.
Degrees of Freedom (df)
For a one-sample t-test, the degrees of freedom are calculated as:
df = n - 1
Degrees of freedom represent the number of independent pieces of information available to estimate a parameter. In this context, it relates to the number of values in a sample that are free to vary after some sample statistic (like the mean) has been calculated.
Interpreting the Results
The calculated t-statistic, along with the degrees of freedom, is used to determine the p-value. The p-value tells you the probability of observing a sample mean as extreme as, or more extreme than, the one you calculated, assuming the null hypothesis (that there is no difference between the sample mean and the hypothesized population mean) is true.
To interpret your results:
- Choose a Significance Level (α): Commonly 0.05 or 0.01. This is your threshold for statistical significance.
- Find the Critical t-Value: Using a t-distribution table or statistical software, find the critical t-value corresponding to your chosen significance level (α) and degrees of freedom (df).
- Compare:
- If the absolute value of your calculated t-statistic is greater than the absolute critical t-value, you reject the null hypothesis. This suggests there is a statistically significant difference between your sample mean and the hypothesized population mean.
- If the absolute value of your calculated t-statistic is less than or equal to the absolute critical t-value, you fail to reject the null hypothesis. This suggests there is not enough evidence to conclude a significant difference.
- Alternatively, use the p-value: If you have access to software that provides the p-value, simply compare it to your significance level (α). If p < α, reject the null hypothesis.
Example Calculation
Let's say a company claims their light bulbs last 1000 hours on average. You test a sample of 25 bulbs and find the following:
- Sample Mean (x̄) = 980 hours
- Hypothesized Population Mean (μ) = 1000 hours
- Sample Standard Deviation (s) = 50 hours
- Sample Size (n) = 25
Using the calculator:
Standard Error = 50 / √25 = 50 / 5 = 10
t = (980 – 1000) / 10 = -20 / 10 = -2.0
Degrees of Freedom (df) = 25 – 1 = 24
With a t-statistic of -2.0 and 24 degrees of freedom, you would then consult a t-distribution table. For a two-tailed test at α = 0.05, the critical t-values are approximately ±2.064. Since |-2.0| < 2.064, you would fail to reject the null hypothesis, meaning there isn't enough evidence to conclude that the bulbs' average lifespan is significantly different from 1000 hours based on this sample.