Ch8 Inference concerning variance Dr. Deshi Ye yedeshi@zju.edu.cn

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

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)

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

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

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

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.

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

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.

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: 1-0.9387=0.0613 > level of significance

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

Testing two variances Null hypothesis

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

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.

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: 1-0.965=0.035 < level of significance