1. Ch8 Inference concerning variance
Dr. Deshi Ye

2. Outline The estimation of variance Hypothesis concerning one variance
Hypothesis concerning two variances

3. One Population Tests One Population Mean Proportion Variance Z Test
t Test
Z Test c 2
Test
(1 & 2
(1 & 2
(1 & 2
(1 & 2
tail)
tail)
tail)
tail)

4. 8.1 The Estimation of Variances
Standard deviation S
Variance
Let
be the sample variance
based on any population having variance

5. Unbiased estimation of a population variance
The sample variance
is an unbiased estimator of
Remark: the sample standard deviation S is not an unbiased estimator of
However, for large samples the bias is small, and it is common practice to estimate with S

6. Confidence interval By Theorem 6.4
is a random variable having the chi-square distribution with n-1 degrees of freedom.
100(1-a)% confidence interval for

7. 8.2 Hypothesis concerning one variance
Consider the problem of testing the null hypothesis that a population variance equals a specified constant against a suitable one-sided or two-sided alternative.
Null Hypothesis
is a random sample from a normal population with the variance is a random variable having the chi-square distribution with n-1 degree of freedom.

8. Criterion Region for testing (Normal population)
Alternative hypothesis
Reject null hypothesis if

9. EX. Testing hypothesis concerning a standard deviation
The lapping process which is used to grind certain silicon wafers to the proper thickness is acceptable only if , the population standard deviation of the thickness of dice cut from the wafers, is at most 0.5 mil. Use the 0.05 level of significance to test the null hypothesis against the alternative hypothesis ,if the thickness of 15 dice cut from such wafers have a standard deviation of 0.64 mil.

10. Solution 1. Null hypothesis: Alternative hypothesis
2. Level of significance: 0.05
3. Criterion: Reject the null hypothesis if
4. Calculation:
5. The null hypothesis cannot be rejected at level 0.05.
6. P-value: = > level of significance

11. 8.3 Hypothesis concerning two variances
If independent random samples of size and are taken from normal population having the same variance, it follows from Theorem 6.5 that
is a random variable having the F distribution with and degrees of freedom

12. Testing two variances
Null hypothesis

13. Criterion Region for testing (Normal population)
Alternative hypothesis
Test statistic
Reject null hypothesis if

14. EX.
It is desired to determine whether there is less variability in the silver plating done by Company 1 than in that done by Company 2. If independent random samples of size 12 of the two companies’ work yield and
, test the null hypothesis against the alternative hypothesis at the 0.05 level of significance.

15. Solution 1. Null hypothesis: Alternative hypothesis
2. Level of significance: 0.05
3. Criterion: Reject the null hypothesis if
4. Calculation:
5. The null hypothesis must be rejected at level 0.05.
6. P-value: =0.035 < level of significance