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## (Basic Statistics for Citizen Data Scientist)

# Power of One Sample Variance Testing

Let represent the hypothetical variance and *s*^{2} the observed variance. Let *x _{+crit}* be the right critical value (based on the null hypothesis with significance level

*α*/2) and

*x*be the left critical value (two-tailed test) , i.e.

_{-crit}*x _{-crit}* = CHIINV(1−

*α*/2,

*n−*1)

*x*= CHIINV(

_{+crit}*α*/2,

*n−*1)

Let *δ* = /*s*^{2}. Then the beta value for the two-tailed test is given by

*β* = CHIDIST(*δ***x _{-crit},n*−1)−CHIDIST(

*δ**

*x*−1)

_{+crit},nFor the one-tailed test H_{0}: ≤ , we use

*x _{crit}*= CHIINV(

*α,n*−1)

*β* = 1−CHIDIST(*δ*x _{crit}*,

*n*−1)

For the one-tailed test H_{0}: ≥ , we use

*x _{crit}* = CHIINV(

*α*,

*n*−1)

*β* = CHIDIST(*x _{crit}*/

*δ,n−*1)

**Example 1**: Calculate the power for the one-tailed and two-tailed tests from Example 3 of One Sample Variance Testing based on a sample of 50 pipes.

The results are shown in Figure 1: 73.4% for the one-tailed test and 63.9% for the two-tailed test.

**Figure 1 – Power of a one sample test of the variance**

**Real Statistics Functions**: The following function is provided in the Real Statistics Resource Pack:

**VAR1_POWER**(*ratio, n, tails, α*) = the power of a one sample chi-square variance test where *ratio* = /*s*^{2} (effect size), *n* = the sample size, *tails* = # of tails: 1 or 2 (default) and *α* = alpha (default = .05).

**VAR1_SIZE**(*ratio, *1−*β, tails, α*) = the minimum sample size required to achieve power of 1−*β* (default .80) in a one sample chi-square variance test where *ratio* = /*s*^{2} (effect size), *tails* = # of tails: 1 or 2 (default) and *α* = alpha (default = .05).

For Example 1, VAR1_POWER(E7,E8,1,E10) = 0.733927 and VAR1_POWER(J7,J8,2,J10) = 0.638379, which is the same as the results shown in Figure 1.

**Example 2**: Calculate the sample size required to find an effect of size .64 with alpha = .05 and power of .80 for the one-tailed and two-tailed tests.

The one tailed test requires a sample size of VAR1_SIZE(.64, .80, 1, .05) = 67 and the two tailed test requires a sample of size VAR1_SIZE(.64) = VAR1_SIZE(.64, .80, 2, .05) = 86.

Statistics for Beginners in Excel – Confidence Intervals for Sampling Distributions

## Statistics for Beginners in Excel – Power of One Sample Variance Testing

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