# 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:

- Population standard deviation: Select the Known radio button if the population standard deviation is known and enter the standard deviation in the provided text field. If it is unknown, select the Unknown radio button.
- If individual samples are entered in a single column of the Datasheet, select the Samples in column: radio button, and select the column name in the drop-down menu.
- To use summary statistics of the sample data, select the Summarized sample data: radio button, and input the sample size, mean, and standard deviation in the provided text fields.
- Enter the confidence level (between 0 and 1) in the Confidence level text field.
- Select the form of the alternative hypothesis in the Alternative Hypothesis: drop-down menu. Enter the hypothesized population mean in the provided text box.
- Click the OK button to perform the computation. The results will be displayed in the log window.

### 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