Statistics > Hypothesis Tests > 1-Population Mean

This utility performs calculations for testing claims about a population mean for the case population standard deviation σ is known and the case when σ is unknown. The population is assumed to be normally distributed. The null hypothesis of a claim about a population mean μ is &mu = &mu0. The alternative hypothesis can be one of the following: &mu < &mu0, &mu > &mu0, or μ ≠ &mu0.

When σ is known, the standard normal distribution is used to calculate p-Values and critical values. The test statistic is computed using the following formula:

where is the sample mean, μ is the hypothesized population mean, and n is the sample size.

When σ is unknown, the Student's t distribution with degrees of freedom n - 1, where n is the sample size, is used to compute p-Values and critical values. The test statistic is computed as follows:

where s is the sample standard deviation.

To use the utility, follow these steps:

Sample Outputs

The null hypothesis and the alternative hypothesis are displayed. The results, along with the input parameters, are displayed in a table.
[N = sample size; Sample Mean = calculated from the individual samples or provided directly by the user; Stdev = s for sample standard deviation, or σ for population standard deviation; Significance Level = 1 - confidence level; Critical Value = critical value corresponding to the significance level; Test Statistic = test statistic corresponding to the hypothesized mean; p-Value = p-Value corresponding to the test statistic]



Hypothesis Test - One Population Mean: confidence level = 0.95
Input: C1
σ unknown
Null hypothesis: μ = 98.6
Alternative hypothesis: μ < 98.6
N Sample Mean Stdev s Significance Level Critical Value Test Statistic p-Value
12 98.392 0.535 0.05 -1.796 -1.349 0.1023

Hypothesis Test - One Population Mean: confidence level = 0.95
Input: C1
σ known
Assumed population standard deviation σ = 0.62
Null hypothesis: μ = 98.6
Alternative hypothesis: μ ≠ 98.6
N Sample Mean Stdev σ Significance Level Critical Value Test Statistic p-Value
12 98.392 0.620 0.05 1.960 -1.164 0.2444