Engineering Statistics Homework Solutions: Confidence Intervals and Hypothesis Testing - P, Assignments of Engineering

Download Engineering Statistics Homework Solutions: Confidence Intervals and Hypothesis Testing - P and more Assignments Engineering in PDF only on Docsity! ENGR 416 HW 4 Solutions 1. From the given data, we have 05.0,83.6,68.12 === αSX . The population mean μ is unknown and the CI has the form XX σα/2 Z± . As specified, we want a 95% confidence interval, the confidence level (1- α) = 0.95, so the critical value is Z α/2=Z0.025 =1.96. We approximate σ with S=6.83, and obtain 9659.050/83.6 ==Xσ . Therefore, the 95% confidence interval is 12.68±1.96*(0.9659), and this is 12.68±1.89 or as (10.79, 14.57). 2. The number of success is X=26 and the number of trials is n=74. We therefore compute 0555.074/)3514.01(*3514.0ˆ/)ˆ1(ˆ3514.074/26ˆ,74ˆ =−=−=== nppandpn For a 90% confidence interval, the value of α/2 is 0.05, so Z α/2 =1.645 (from Appendix Table). The 90% confidence interval is therefore 0555.0*645.13514.0 ± or (0.2601, 0.4427). 3. 99% confidence interval on mean current required Assume that the data are a random sample from a normal distribution. 7.152.31710 === sxn 250.39,005.0 =t 34.33306.301 10 7.15250.32.317 10 7.15250.32.317 9,005.09,005.0 ≤≤ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +≤≤⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +≤≤⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − μ μ μ n stx n stx 4. 95% confidence interval for σ: given n = 51, s = 0.37 First find the confidence interval for σ2 : For α = 0.05 and n = 51, 71.42 and 32.36 χα / ,2 12 n− = χ0 025 502. , = χ α1 2 12− − =/ ,n χ0 975 502. , = 36.32 )37.0(50 42.71 )37.0(50 222 ≤≤ σ 0.096 ≤ σ2 ≤ 0.2115 Taking the square root of the endpoints of this interval we obtain, 0.31 < σ < 0.46 5. The worst case would be for p = 0.5, thus with E = 0.05 and α = 0.01, zα/2 = z0.005 = 2.58 we obtain a sample size of: 64.665)5.01(5.0 05.0 58.2)1( 22 2/ =−⎟ ⎠ ⎞ ⎜ ⎝ ⎛=−⎟ ⎠ ⎞ ⎜ ⎝ ⎛= pp E zn α , n ≅ 666 1