Statistical Methods - Practice Test 3 (Winter 2013) 1. The sample information given in the table below contains a summary of the weights (in lbs) of 15 randomly selected women and 13 randomly selected men from a certain city. Group female male Sample Size 15 13 Sample Mean 132.05 177.29 Sample Standard Deviation 5.54 4.88 (a) Test the hypothesis that the variance in men’s weight is different from 25. Use a 5% significance level and the P -value approach. (b) Construct a 90% confidence interval for the true variance in the weight of women. (c) Construct a 90% confidence interval for the difference in the average weight of women and men. (d) Test the hypothesis that men are on average at least 40 pounds heavier than women. Use a 1% significance level and the rejection region approach. (e) Perform a hypothesis test, at the 10% significance level, to determine if there is a significant difference in the weigth variances among females and males. Use the rejection region approach. (f) Interpret in words your conclusions from the previous parts. 2. Is iron deficiency more common among vegans than among omnivores? A sample of 122 adult male vegans found that 31 suffered from iron deficiency while a sample of 257 adult male omnivores found that 63 suffered from iron deficiency. After clearly identifying the relevant hypotheses, is there enough evidence to conclude that iron deficiency is more common among vegans than among omnivores (at the 5% significance level)? Use the P -value approach. 3. An investigation was carried out to study the relationship between speed (in m/sec.) and stride rate (number of steps taken per second) among female marathon runners. Resulting summary quantities included n = 11, Σ (speed)= 205.4, Σ (speed)2 = 3880.08, Σ (rate)= 35.16, Σ (rate)2 = 112.681, and Σ (speed)(rate)= 660.130. (a) Calculate the equation of the regression line that you would use to predict stride rate from speed. (b) Calculate the equation of the regression line that you would use to predict speed from stride rate. (c) For a runner with a speed of 15 meters per second, what is the probability that she has a stride exceeding 3 steps per second? 4. Toughness and fibrousness are major determinants of quality in asparagus. Letting x = shear force (in kg) and y = percent fiber dry weight, you are given the summary quantities: n = 18, Σxi = 1950, Σx2i = 251 970, Σyi = 47.92, Σyi2 = 130.6074, and Σxi yi = 5100.92. (a) Calculate the value of the sample correlation coefficient. Does it indicate strong linear correlation between the variables? (b) Carry out a test at the 1% significance level to decide whether there is a negative linear correlation between the two variables. 5. Consider a hypothesis test on a mean, with H0 : µ = 8 and the alternative Ha : µ 6= 8, at the 5% significance level. Assume the sample size is 60 and the population standard deviation is σ = 6. (a) Draw the rejection region for this test. (b) Calculate the probability of a type II error when the true population average is µ = 5. Start from the definition of the type II error, and show all the steps. (c) Without calculating it, would β(2) be bigger or smaller than β(5)? 6. Trace metals in drinking water affect the flavor and an unusually high concentration can pose a health hazard. Ten pairs of data were taken measuring zinc concentration in bottom water and surface water. Does the data suggest that the true average concentration in the bottom water exceeds that of surface water? Use a 1% significance level and the rejection region approach. Zinc concentration at bottom Zinc concentration on surface 1 .430 .415 2 .266 .238 3 .567 .390 4 .531 .410 5 .707 .605 6 .716 .609 7 .651 .632 8 .589 .523 9 .469 .411 10 .723 .612 7. The management of a large corporation is conducting a study of employee satisfaction under three work schedules: • a 5-day work week with 8 hour shifts per day, • a 4-day work week with 10 hour shifts per day, • a 3-day work week with 13 hour shifts per day. A random sample of 400 employees were assigned one of the schedules. After six months, those employees were surveyed regarding their satisfaction level with the schedule; results are shown in the table below. Determine if the work schedule and satisfaction level are independent. Use a 1% significance level and the rejection region approach. In addition, also estimate the P -value. Satisfaction level High Medium Low 5-day schedule 65 45 40 4-day schedule 86 25 39 3-day schedule 60 15 25 8. For each situation described below, find or estimate the P -value using the distribution tables. (a) n = 60 Ha : µ < 40 z ∗ = −1.24 (b) n = 12 Ha : µ 6= 12 t∗ = 2.181 (c) n = 35 Ha : σ 2 < 0.5 χ2,∗ = 15.5 (d) n = 29 Ha : σ 2 > 1 χ2,∗ = 15.5 (e) n1 = 11, n2 = 21 Ha : σ12 > σ22 f ∗ = 3.01 (f) n1 = 45, n2 = 50 Ha : p1 − p2 6= 0 z ∗ = −2.01 ANSWERS: 1. (a) χ2 = 11.4309, P > 0.2, fail to reject H0 (b) (18.1415, 65.3609) (c) (41.8765, 48.6035) (d) ν = 25, t∗ = 2.6609, tc = 2.485, (e) f ∗ = 1.2888, 2. z ∗ = 0.1888, reject H0 fail to reject H0 P = 0.4247, fail to reject H0 3. (a) Y = 1.6932 + 0.0805X (b) Y = −20.0509 + 12.1149X (c) 0.003 4. (a) r = −0.2572 (b) tc = −2.583, t∗ = −1.0646, fail to reject H0 5. (a) z < −1.96 and z > 1.96 (b) β(5) = Φ(5.83) − Φ(1.91) ≈ 0.0281 (c) β(2) is smaller than β(5) 6. d = 0.0804, 7. ν = 4, s = 0.0523, χ2,∗ = 12.9049, t∗ = 4.8638, reject H0 0.01 < P < 0.025, fail to reject H0 8. (a) P ≈ 0.1075 (b) 0.05 < P < 0.1 (c) P < 0.005 (d) 0.95 < P < 0.975 (e) 0.01 < P < 0.05 (f) P ≈ 0.0444 (or if using Table A.8, 0.05 < P < 0.06)