Hypothesis Testing - Statistical Test of Variance

Hypothesis Testing - Statistical Test of Variance - Theory & Examples

Statistical Hypothesis: Testing Variance -- Population

Population distribution:

Population distribution: X is normally distributed, variance is unknown, expected value is known or unknown

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Statistical Hypothesis: Testing Variance -- Sample

Sample statistics:

sample statistic: n times the sample variance over the population variance

Sample distribution:

the sample statistic has a chi-square distribution with n degrees of freedom (mean is known) or n-1 degrees of freedom (mean is unknown)

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Statistical Hypothesis: Testing Variance -- Critical Region

Table overview:

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Statistical Hypothesis: Testing Variance -- the Chi-square distribution

Definition

Let X be a stochastic variable following a normal distribution with expected value mu and variance equal to sigma-squared

From this it follows that u = (X - mu) over sigma is standard normally distributed

The Chi-square distribution with one degree of freedom is defined as the square of a standard normal distributed variate

The Chi-square distribution with n degrees of freedom is defined as the sum of n squared in-dependent standard normal distributed variates

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Property 1

The sum (difference) of two independent Chi-square distributed variates, with degrees of freedom n1 and n2 respectively, is also Chi-square distributed with degrees of freedom equal to the sum (difference) of the degrees of freedom (n1, n2).

Property 2

The expected value of a Chi-square distributed variate is equal to the number of degrees of freedom

The variance of a Chi-square distributed variate is equal to two times the number of degrees of freedom

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Statistical Hypothesis: Testing Variance -- Approximation of the Chi-square distribution

Rule of thumb

For larger samples, i.e. for sample sizes n > 30, the distribution can be approximated by the standard normal distribution.

Example

Find the critical value for the chi-square distribution if n = 30 and alpha (type I error) = 5%

using the normal approximation

since k = 1.645 it follows that the approximation results in c = 43.49 against a correct tabulated value of 43.773.

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Statistical Hypothesis: Testing Variance -- Distribution of Sample Variance

Proof

formula

solution

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From a random sample, with sample size n and drawn from a population following a normal distribution and given mean and standard deviation, the sample variance can be estimated as described in the following cases.

Estimation - Case 1: mean is unknown

formula

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Estimation - Case 1: mean is known

formula

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Statistical Hypothesis: Testing Variance -- Summary

Estimation of variance - distribution of test statistic - degrees of freedom

summary

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Statistical Hypothesis: Testing Variance -- Example 1: Critical Value (Region)

Free Statistics Software (Calculator)

Problem

problem

Solution

solution

Conclusion

conclusion

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Statistical Hypothesis: Testing Variance -- Example 2: P-value (probability)

Free Statistics Software (Calculator)

Problem

problem

Solution

solution

Conclusion

conclusion

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Statistical Hypothesis: Testing Variance -- Example 3: Confidence intervals for Sample Variance

Free Statistics Software (Calculator)

Problem

problem

Solution

solution (part 1)

solution (part 2)

solution (part 3)

Conclusion

conclusion

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Statistical Hypothesis: Testing Variance -- Example 4: Confidence intervals for Population Variance

Free Statistics Software (Calculator)

Problem

problem

Solution

solution (part 1)

solution (part 2)

solution (part 3)

Conclusion

conclusion

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