One sample t-test calculator

The this one-sample t-test calculator helps you determine whether the mean of a single sample is significantly different from a known or hypothesized population mean.

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What is a one-sample t-test?

A one-sample t-test is a statistical test used to determine whether the mean of a single sample is significantly different from a known or hypothesized population mean. It is commonly used when the population standard deviation is unknown, and the sample size is small (typically n < 30), but it can also be used for larger samples.

One-sample t-test formula

The test statistic (t-statistic) is calculated as:

t = \frac{\bar{x} - \mu}{s / \sqrt{n}}

where:

• \bar{x} = sample mean
• \mu = hypothesized population mean
• s = sample standard deviation
• n = sample size

The resulting t-statistic follows a t-distribution with n−1 degrees of freedom (df).

Example of a one-sample t-test

Scenario

A factory claims that the average weight of a product is 500 grams. A quality control inspector randomly selects 25 products and finds an average weight of 495 grams with a standard deviation of 10 grams. The inspector wants to test if the average weight is significantly different from 500 grams at a 5% significance level \alpha = 0.05).

Step 1: Define hypotheses

Null hypothesis (H_0​): The true mean is 500 grams

H_0: \mu = 500

Alternative hypothesis (H_a​): The true mean is not 500 grams

H_a: \mu \neq 500

Step 2: Calculate the t-Statistic

Using the formula:

t = \frac{495 - 500}{10 / \sqrt{25}}

t = \frac{-5}{2} = -2.5

Step 3: Find the critical t-Value

Since n=25, the degrees of freedom is:

df=25−1=24

Using a t-table, the critical t-value for a two-tailed test at \alpha = 0.05 and df = 24 is ±2.064.

Step 4: Compare t-Statistic with critical value

The calculated t=−2.5 is beyond −2.064, meaning we reject the null hypothesis.

Step 5: Conclusion

Since t=−2.5 is less than −2.064, we conclude that the average weight of the product is significantly different from 500 grams at the 5% significance level.

Interpreting p-Values

Alternatively, we can use a p-value approach:

• Using a p-value calcualtor from t-statistic, the p-value for t=−2.5 with 24 degrees of freedom is 0.020.
• Since p<0.05, we reject H_0​ and conclude that the average weight is significantly different from 500g.

When to use a one-sample t-test?

Use a one-sample t-test when:

• You have one sample.
• You do not know the population standard deviation.
• The sample is randomly selected and follows a normal distribution (or nnn is large enough for the Central Limit Theorem to apply).

Summary