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Hypothesis Testing - Statistical Test of Population Mean with unknown Variance - Theory & Examples

[Home] [Introduction] [Mean (var. known)] [Variance] [Mean (var. unknown)]

Theory: [Case 1] [Case 2]
Practice: [Software (Calculator)]



Statistical Hypothesis: Testing Mean with unknown Variance -- Case1

Assume that

U has a standard normal distribution, and V has a Chi-square distribution with n-1 degrees of freedom

with

the mean of x is normally distributed

and

sample variance and mean

and assume that U and V are independent.

The t-density is defined as follows

definition of t-density

Therefore

proof that the ratio of U and V is t-distributed

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Statistical Hypothesis: Testing Mean with unknown Variance -- Case 2

Assume that

U has a standard normal distribution, and V has a Chi-square distribution with n-1 degrees of freedom

with

the mean of x is normally distributed

and

sample variance and mean

and assume that U and V are independent.

The t-density is defined as follows

definition of t-density

Therefore

proof that the ratio of U and V is t-distributed

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Statistical Hypothesis: Testing Mean with unknown Variance -- Free Statistics Software (Calculator)

Confidence Interval - Univariate Dataset

use this Software (Calculator) to perform statistical hypothesis testing about the mean when de population variance is unknown

P-value - Univariate Dataset

use this Software (Calculator) to perform statistical hypothesis testing about the mean when de population variance is unknown

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