2Outline The estimation of variance Hypothesis concerning one variance Hypothesis concerning two variances
3One Population Tests One Population Mean Proportion Variance Z Test t TestZ Testc2Test(1 & 2(1 & 2(1 & 2(1 & 2tail)tail)tail)tail)
48.1 The Estimation of Variances Standard deviation SVarianceLetbe the sample variancebased on any population having variance
5Unbiased estimation of a population variance The sample varianceis an unbiased estimator ofRemark: the sample standard deviation S is not an unbiased estimator ofHowever, for large samples the bias is small, and it is common practice to estimate with S
6Confidence 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
78.2 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 Hypothesisis 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.
8Criterion Region for testing (Normal population) Alternative hypothesisReject null hypothesis if
9EX. 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.
10Solution 1. Null hypothesis: Alternative hypothesis 2. Level of significance: 0.053. Criterion: Reject the null hypothesis if4. Calculation:5. The null hypothesis cannot be rejected at level 0.05.6. P-value: = > level of significance
118.3 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 thatis a random variable having the F distribution with and degrees of freedom
13Criterion Region for testing (Normal population) Alternative hypothesisTest statisticReject null hypothesis if
14EX.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.
15Solution 1. Null hypothesis: Alternative hypothesis 2. Level of significance: 0.053. Criterion: Reject the null hypothesis if4. Calculation:5. The null hypothesis must be rejected at level 0.05.6. P-value: =0.035 < level of significance