# TEST_MEAN - Population Mean Test

Calculates the p-value of the statistical test for the population mean.

## Syntax

TEST_MEAN(x, mean, Return_type, Alpha)

x
is the input data sample (one/two-dimensional array of cells (e.g., rows or columns))
mean
is the assumed population mean. If missing, the default value of zero is assumed.
Return_type
is a switch to select the return output (1 = P-Value (default), 2 = Test Stats, 3 = Critical Value).
Method Description
1 P-Value.
2 Test Statistics (e.g. Z-score).
3 Critical Value.
Alpha
is the statistical significance of the test (i.e., alpha). If missing or omitted, an alpha value of 5% is assumed.

## Remarks

1. The sample data may include missing values (e.g., #N/A).
2. The test hypothesis for the population mean: $$H_{o}: \mu=\mu_o$$ $$H_{1}: \mu\neq \mu_o$$ Where:
• $H_{o}$ is the null hypothesis.
• $H_{1}$ is the alternate hypothesis.
• $\mu_o$ is the assumed population mean.
• $\mu$ is the actual population mean.
3. For the case in which the underlying population distribution is normal, the sample mean/average has a Student's t with T-1 degrees of freedom sampling distribution: $$\bar x \sim t_{\nu=T-1}(\mu,\frac{S^2}{T})$$ Where:
• $\bar x$ is the sample average.
• $\mu$ is the population mean/average.
• $S$ is the sample standard deviation. $$S^2 = \frac{\sum_{i=1}^T(x_i-\bar x)^2}{T-1}$$
• $T$ is the number of non-missing values in the data sample.
• $t_{\nu}()$ is the Student's t-Distribution.
• $\nu$ is the degrees of freedom of the Student's t-Distribution.
4. The Student's t-Test for the population mean can be used for small and for large data samples.
5. This is a two-sides (i.e., two-tails) test, so the computed p-value should be compared with half of the significance level ($\alpha/2$).
6. The underlying population distribution is assumed normal (gaussian).