Introduction
The estimation and tests presented here are based on the a priori existence of a target or reference value, and whether the population mean differs from that value or not. However, it is assumed that the population variance is unknown. For the tests, t-tests are used.
Example
Estimation and testing of the population mean when there is only one population (population variance is unknown)
\[X:32.3,27.0,30.5,36.5,36.2,30.7,30.5,31.0,24.3,27.7\]
Point Estimation of Mother Mean
Point Estimation:
\[\bar{x}=30.67\]
Interval estimation with 95% confidence rate
Interval estimation:
\[(\bar{x}-2.262\sqrt{\frac{var1}{n}},\bar{x}+2.262\sqrt{\frac{var1}{n}})=(27.95,33.39)\]
\[var1=\frac{Σ_{i=1}^n(x_i-\bar{x})}{n-1}\]
rejection zone
\[|t_0|≧t(9,0.05)=2.262\]
Test statistics, standardization, and
\[t_0=\frac{\bar{x}-μ_0}{\sqrt{\frac{var1}{n}}}=0.557\]
It cannot be said that the population mean is not \(μ_0\)=30.
MAXIMA function for t-test
Function: test_mean (x)
Function: test_mean (x, option_1, option_2, ...)
This is the mean t-test. Argument x is a list or a column matrix containing a one dimensional sample. It also performs an asymptotic test based on the Central Limit Theorem if option 'asymptotic is true.
Options:
- 'mean, default 0, is the mean value to be checked.
- 'alternative, default 'twosided, is the alternative hypothesis; valid values are: 'twosided, 'greater and 'less.
- 'dev, default 'unknown, this is the value of the standard deviation when it is known; valid values are: 'unknown or a positive expression.
- 'conflevel, default 95/100, confidence level for the confidence interval; it must be an expression which takes a value in (0,1).
- 'asymptotic, default false, indicates whether it performs an exact t-test or an asymptotic one based on the Central Limit Theorem; valid values are true and false.
The output of function test_mean is an inference_result Maxima object showing the following results:
- 'mean_estimate: the sample mean.
- 'conf_level: confidence level selected by the user.
- 'conf_interval: confidence interval for the population mean.
- 'method: inference procedure.
- 'hypotheses: null and alternative hypotheses to be tested.
- 'statistic: value of the sample statistic used for testing the null hypothesis.
- 'distribution: distribution of the sample statistic, together with its parameter(s).
- 'p_value: p-value of the test.
