# 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).

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11
A B
Date Data
1/1/2008 #N/A
1/2/2008 -0.95
1/3/2008 -0.88
1/4/2008 1.21
1/5/2008 -1.67
1/6/2008 0.83
1/7/2008 -0.27
1/8/2008 1.36
1/9/2008 -0.34
1/10/2008 0.48

Formula Description (Result)
=AVERAGE($B$2:$B$11) Sample mean (-0.0256)
=TEST_MEAN($B$2:$B$11,0) p-value of the test (0.472)

## 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
• Hurst, Simon, The Characteristic Function of the Student-t Distribution , Financial Mathematics Research Report No. FMRR006-95, Statistics Research Report No. SRR044-95