Inferences about Population Variance Lecture 35

 Inferences about Population Variance

When several samples are available

Lecture 35

Confidence Interval for Population Variance when several Samples are Available

Let S1, S2,..., Sk be the estimates of population standard deviation "σ." computed from the values of k random independent samples of sizes n1, n2,..., nk selected from a normal population having mean "μ" and standard deviation "σ." 

The pooled estimate of population variance is given by:

Where:

n = n1 + n2 + ...+ nk

The sampled population is normal, so the sampling distribution of ∑(niSi2) / σ2 approaches the chi square distribution to (n - k) degrees of freedom.

A 100 (1-α)%  confidence interval for population variance is given by:

Example 9.6: Three independent random samples, each of size 8, gave the sample variances 3.80, 3.20, and 1.25. Obtained the pooled estimate of population variance and also used it to find the 95% confidence interval for population variance.

Solution:

n1 = n2 = n3 = 8

S1^2 = 3.80, S2^2 = 3.20, S3^2 = 1.25

The pooled estimate of population variance is given by:

1-α=0.95

α = 0.05

χ2α/2 (n-k) = χ20.025(21) = 35.479

χ21-α/2 (n-k) = χ20.975(21) = 10.283

Example 9.10: The following data from a normal population with mean "μ" and standard deviation "σ."

X1 = 4, 6, 5, 7; X2 = 7, 3, 6; X3 = 5, 3, 7, 9. Find the pooled estimate of population variance and construct a 98% confidence interval for population variance.

Solution:

1-α=0.98

α = 0.02

χ2α/2 (n-k) = χ20.01(8) = 20.090

χ21-α/2 (n-k) = χ20.99(8) = 1.646

Testing Hypothesis about the Equality of Variances of k (k > 2) Normal Populations 

(M.S Bartlett Test)

The Bartlett's test is used to test the homogenity of normal population variances. 

Several test procedures are developed to test the above null hypothesis. The M.S. Bartlett is one of them presented here. This test procedure is based on a statistic “u” whose sampling distribution is approximately a chi square distribution with (k-1) degree of freedom.

Testing Procedure:

i. State null & alternative hypothesis

ii. The significance level: α

iii. The test statistic: The sampled population is normal.

iv. Critical Region:

Reject H0 if χ2 > χ2α (k-1) 

v. Computation:

vi. Remarks

Example 9.11: The four independent samples selected from four normal populations gave the following results:

Sample 1	Sample 2	Sample 3	Sample 4
150	110	150	140
160	130	130	170
130	180	120	100
170	160	160	120

Use Bartlett's test to test the hypothesis of equal variance at the 5% significance level.

Solution:

i. State the null & alternative hypotheses

ii. The significance level: α

iii. The test statistic: The sampled population is normal.

iv. Critical Region:

Reject H0 if χ2 > χ2 0.05 (4-1) = 7.815

v. Computation:

 

vi. Remarks: The chi square calculated value falls in the rejection region; the sample data does not provide sufficient evidence to accept the null hypothesis. Thus, it is concluded the population variances are not homogenous. 

• Read More: Hypothesis Testing about Equality of Several Population Proportions