IT233: Applied Statistics TIHE 2005 Confidence Intervals and Tests of Hypotheses Confidence intervals and tests for Confidence intervals and tests for 1 2 Confidence intervals and tests for 2 Confidence intervals and tests for 2 / 2 1 2 Confidence Intervals and Tests Concerning σ 2 Confidence Intervals for σ2 Theorem: If S 2 is the variance of a random sample of size n from a normal population, a 1 100% confidence interval for 2 is n 1 S 2 2 n 1 S 2 2 where 2 and 2 1 2 leaving areas 2 1 2 2 are 2 - values with n 1 degrees of freedom, 2 2 and 1 2 , respectively, to the right. Proof: The confidence interval for 2 can be derived by using the statistic n 1 S 2 2 2 1 Consider the following figure: 2 2 1 2 1 0 2 2 2 2 To derive the confidence interval, we begin by giving a probability statement based on the above figure that P 12 2 2 1 2 2 Substituting for 2 , we write n 1 S 2 2 2 P 1 1 2 2 2 To isolate 2 in the centre of the inequality, we divide each term by (n 1)S 2 and then inverting each term, we obtain 2 n 1 S 2 2 n 1 S P 2 2 1 1 2 2 The desired 1 100% confidence interval for 2 can be read as: n 1 S 2 2 n 1 S 2 2 2 2 1 2 2 Example 1: A manufacturer of batteries claims that his batteries will last, on average, 3 years with a variance of 1 year. If 5 of these batteries have lifetimes 1.9, 2.4, 3.0, 3.5, 4.2 years construct a 95% C.I. for 2 and decide if the claim that 2 1 is valid. Solution: First we find the summary data: NOTE: xi2 = 48.26 xi = 15, n = 5, n xi2 xi 2 S n n 1 2 5 48.26 152 = 0.815 5 4 You could use your Calculator to find: S 2 n 1 We have, 2 0.902773504 2 0.815 0.05 n 1 4 2 2 0.025 11.143 2 2 1 2 2 0.975 0.484 the 95% Confidence Interval for 2 is n 1 S 2 2 n 1 S 2 2 2 2 1 2 4 0.815 4 0.815 2 11.143 0.484 0.293 2 6.736 3 To answer the last part of the problem, that is, “Is the claim that 2 1 valid?”, we use the confidence interval method. Since above interval reveals that the value 2 1 lies in the internal, we can accept the claim with a 95% confidence. Answer: Notes: (1) Do not use confidence interval method for testing any claim, if the test is one-tailed. (2) You could also use the following traditional method. Test of Hypothesis for 2 The traditional method of hypothesis testing for the above example is as follows: Null hypothesis: H : 2 1 0 Alternative hypothesis: H : 2 1 (Two-tailed test) 1 Critical Region: 0.05 n 1 4 2 =0.025 2 2 0.025 11.143 2 = 0.025 2 2 2 0.975 0.484 1 0 Acceptance Region 0.484 11.1434 2 the null hypothesis is rejected when 2 0.484 and 2 11.143. 4 Test Statistic 2 n 1 S 2 2 4 0.815 1 3.26 Conclusion: Since the value of test statistic 2 3.26 falls in the acceptance region, we accept H0 : 2 1 . 5