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

- The sample data may include missing values (e.g., #N/A).
- 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.

- 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.

- The Student's t-Test for the population mean can be used for small and for large data samples.
- This is a two-sides (i.e., two-tails) test, so the computed p-value should be compared with half of the significance level ($\alpha/2$).
- The underlying population distribution is assumed normal (gaussian).

## Files Examples

## Related Links

- Wikipedia - Statistical hypothesis testing.
- Wikipedia - Student's t-test.
- Wikipedia - Student's t-distribution.

## References

- George Casella; Statistical Inference; Thomson Press (India) Ltd; (Dec 01, 2008), ISBN: 8131503941.
- K.L. Lange, R.J.A. Little, and J.M.G. Taylor. "Robust Statistical Modeling Using the t Distribution." Journal of the American Statistical Association 84, 881-896, 1989.

## Comments

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