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Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com 8.8 Small-Sample Confidence Intervals for µ and µ 1 − µ 2 427 Find a 95% confidence interval for the true average velocity µ for shells of this type. Assume that muzzle velocities are approximately normally distributed. Solution If we assume that the velocities Y i are normally distributed, the confidence interval for µ is Y ± t α/2 ( S √n ), where t α/2 is determined for n−1 df. For the given data, y = 2959 and s = 39.1. In this example, we have n − 1 = 7 df and, using Table 5, Appendix 3, t α/2 = t .025 = 2.365. Thus, we obtain ( 39.1 2959 ± 2.365 √ ), or 2959 ± 32.7, 8 as the observed confidence interval for µ. Suppose that we are interested in comparing the means of two normal populations, one with mean µ 1 and variance σ 2 1 and the other with mean µ 2 and variance σ 2 2 .Ifthe samples are independent, confidence intervals for µ 1 − µ 2 based on a t-distributed random variable can be constructed if we assume that the two populations have a common but unknown variance, σ 2 1 = σ 2 2 = σ 2 (unknown). If Y 1 and Y 2 are the respective sample means obtained from independent random samples from normal populations, the large-sample confidence interval for (µ 1 − µ 2 ) is developed by using Z = (Y 1 − Y 2 ) − (µ 1 − µ 2 ) √ σ 2 1 n 1 + σ 2 2 n 2 as a pivotal quantity. Because we assumed that the sampled populations are both normally distributed, Z has a standard normal distribution, and using the assumption σ 2 1 = σ 2 2 = σ 2 , the quantity Z may be rewritten as Z = (Y 1 − Y 2 ) − (µ 1 − µ 2 ) √ 1 + 1 . n 1 n 2 σ Because σ is unknown, we need to find an estimator of the common variance σ 2 so that we can construct a quantity with a t distribution. Let Y 11 , Y 12 , ..., Y 1n1 denote the random sample of size n 1 from the first population and let Y 21 , Y 22 , ..., Y 2n2 denote an independent random sample of size n 2 from the second population. Then Y 1 = 1 ∑n 1 Y 1i and Y 2 = 1 ∑n 2 Y 2i . n 1 n 2 i=1 i=1

www.downloadslide.com 428 Chapter 8 Estimation The usual unbiased estimator of the common variance σ 2 is obtained by pooling the sample data to obtain the pooled estimator S 2 p : S 2 p = ∑ n1 i=1 (Y 1i − Y 1 ) 2 + ∑ n 2 i=1 (Y 2i − Y 2 ) 2 n 1 + n 2 − 2 = (n 1 − 1)S1 2 + (n 2 − 1)S2 2 , n 1 + n 2 − 2 where Si 2 is the sample variance from the ith sample, i = 1, 2. Notice that if n 1 = n 2 , S 2 p is simply the average of S2 1 and S2 2 .Ifn 1 ≠ n 2 , S 2 p is the weighted average of S2 1 and S2 2 , with larger weight given to the sample variance associated with the larger sample size. Further, W = (n 1 + n 2 − 2)S 2 ∑ n1 p i=1 = (Y 1i − Y 1 ) 2 ∑ n2 i=1 + (Y 2i − Y 2 ) 2 σ 2 σ 2 σ 2 is the sum of two independent χ 2 -distributed random variables with (n 1 − 1) and (n 2 −1) df, respectively. Thus, W has a χ 2 distribution with ν = (n 1 −1) + (n 2 −1) = (n 1 + n 2 − 2) df. (See Theorems 7.2 and 7.3.) We now use the χ 2 -distributed variable W and the independent standard normal quantity Z defined in the previous paragraph to form a pivotal quantity: T = Z √ W ν ⎡ ⎤ = ⎢ (Y 1 − Y 2 ) − (µ 1 − µ 2 ) ⎣ √ ⎥ 1 σ + 1 ⎦ n 1 n 2 /√ (n 1 + n 2 − 2)S 2 p σ 2 (n 1 + n 2 − 2) = (Y 1 − Y 2 ) − (µ 1 − µ 2 ) S p √ 1 n 1 + 1 n 2 , a quantity that by construction has a t distribution with (n 1 + n 2 − 2) df. Proceeding as we did earlier in this section, we see that the confidence interval for (µ 1 − µ 2 ) has the form √ 1 (Y 1 − Y 2 ) ± t α/2 S p + 1 , n 1 n 2 where t α/2 is determined from the t distribution with (n 1 + n 2 − 2) df. EXAMPLE 8.12 To reach maximum efficiency in performing an assembly operation in a manufacturing plant, new employees require approximately a 1-month training period. A new method of training was suggested, and a test was conducted to compare the new method with the standard procedure. Two groups of nine new employees each were trained for a period of 3 weeks, one group using the new method and the other following the standard training procedure. The length of time (in minutes)

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