Download Stat 345 Solutions: Confidence Intervals and One-Sample T-Tests and more Assignments Mathematical Statistics in PDF only on Docsity! Stat 345 Solutions - Section 8.3 Problem 8-20 This is just a direct lookup in the table, giving values of (a) 2.179, (b) 2.064, (c) 3.012, (d) 3.733 Problem 8-22 Since σ is unknown and the sample size is small, we will assume that the distribution of tire life is normal. Then, the general form of the confidence interval will be (x̄− tα/2,n−1 σ√ n , x̄ + tα/2,n−1 σ√ n ) Here, x̄ = 60, 139.7, s = 3645.94, n = 16, and t0.05/2,15 = 2.131 since we want to construct a 95% CI. Thus, we have (60139.7− (2.131)3645.94 4 , 60139.7 + (2.131) 3645.94 4 ) (58, 197.33 , 62, 082.07). Problem 8-23 A one sided CI puts all of α in one tail, so compute x̄− t0.1s/ √ n = 1.25− 2.539(0.25)/√20 Problem 8-24 With the small n we need to assume the population we sampled from (microamp levels of all TV tubes of this type) is normally distributed. Without specifying otherwise, CI’s are two-sided, so compute x̄± t0.005s/ √ n = 317.2± 3.250(15.7)/√10 Problem 8-25 (b) Since σ is unknown and the sample size is small, we will assume that the distribution of polyunsaturated fatty acid level is normal. Then, the general form of the confidence interval will be (x̄− tα/2,n−1 σ√ n , x̄ + tα/2,n−1 σ√ n ) Here, x̄ = 16.98, s = 0.1017, n = 6, and t0.01/2,5 = 4.032 since we want to construct a 99% CI. Thus, we have (16.98− (4.032)0.1017√ 6 , 16.98 + (4.032) 0.1017√ 6 ) (16.81 , 17.15). We are 99% confident that this interval covers the true mean level of polyunsaturated fatty acid. i.e. In order to get data like this we must have sampled from a population with a mean in this range. We are right 99% of the time when we make such claims. 8.26 (a) There is nothing in this boxplot to suggest a problem with assuming we sampled from a normal distribution. We have near-perfect symmetry (more than we need to support the assumption). Co m p. S tr en gt h 2320 2300 2280 2260 2240 2220 2200 Boxplot of Comp. Strength (b) One-Sample T: Comp. Strength Variable N Mean StDev SE Mean 95% CI Comp. Strength 12 2259.92 35.57 10.27 (2237.32, 2282.52) (c) One-Sample T: Comp. Strength 95% Lower Variable N Mean StDev SE Mean Bound Comp. Strength 12 2259.92 35.57 10.27 2241.48 8.27 (a) This boxplot does not show perfect symmetry, but there are no outliers or other reasons to worry much about assuming we sampled from a normal distribution. R od D ia m et er 8.28 8.26 8.24 8.22 8.20 Boxplot of Rod Diameter (b) One-Sample T: Rod Diameter Variable N Mean StDev SE Mean 95% CI Rod Diameter 15 8.23400 0.02530 0.00653 (8.21999, 8.24801)
