Final Examination in Statistics: Problem Solving and Hypothesis Testing - Prof. Cecile M. , Exams of Data Analysis & Statistical Methods

Download Final Examination in Statistics: Problem Solving and Hypothesis Testing - Prof. Cecile M. and more Exams Data Analysis & Statistical Methods in PDF only on Docsity! Stat./For./Hort. 571 Larget and Zhu December 21, 2005 Final Examination Name: Please indicate the sections that you attend. Lecture: (circle one) Bret Larget Jun Zhu Discussion: (circle one) Sang-Hoon Cho Xiwen Ma Tao Yu Instructions: 1. The exam is open book. You may use the Course Notes, other texts, lecture notes, homework solutions, your notes, and a calculator. You may not use a laptop computer. 2. Do all your work in the spaces provided. If you need additional space, use the back of the preceding page, indicating clearly that you have done so. 3. To receive full credit, you must show your work. We will award partial credit. 4. Use your time wisely. Do not dwell too long on any one question. Answer as many questions as you can in the time allowed. 5. Note that some questions have multiple parts. For some questions, these parts are independent, so you can work on part (b) or (c) separately from part (a). For graders’ use. Question Possible Points Score 1 20 2 20 3 20 4 20 5 20 Total 100 1. Weights of ears of organic sweet corn grown are approximately normally distributed with a mean of 11.1 ounces and a standard deviation of 3.1 ounces. (a) The lightest 8% of the ears all weigh less than what weight? (b) What is the probability that a single ear of corn chosen at random from this population weighs more than 12.0 ounces? (c) What is the probability that the mean weight of a random sample of a dozen ears of corn weighs more than 12.0 ounces? (d) In a random sample of a dozen ears of corn, what is the probability that one or fewer ears of corn weigh more than 12.0 ounces? (e) In a random sample of 288 ears of corn, use an approximation method to estimate the probability that 125 or more ears weigh more than 12.0 ounces. 4. Continue the previous problem with the following additional information. Ground water near the industrial plant flows in a known direction. Upgradient wells A and B are located in an area that is the source of ground water near the industrial plant. In contrast, downgradient wells C, D, E, and F are located in an area where much of the ground water has passed through the location of the industrial plant. Four specific null hypotheses of interest are as follows. 1. H0 : µC − (µA + µB)/2 = 0 2. H0 : µD − (µA + µB)/2 = 0 3. H0 : µE − (µA + µB)/2 = 0 4. H0 : µF − (µA + µB)/2 = 0 (a) Use the Bonferroni method to test the hypotheses simultaneously versus one-sided alternative that the difference is positive with an experiment-wide error rate of α = 0.04 (please note the unconventional choice). (b) Summarize your conclusions in the context of the problem. (c) Briefly explain how the particular tested contrasts address questions of practical and scientific importance. 5. Researchers are interested in studying the change in bone density as adults age. To the right is a scatter plot of Y , lumbar spine bone density (grams per cm3), versus X, age (years), in a sample of 41 American women. Some useful summary statistics are below. x̄ = 48.9, ȳ = 0.759, ∑ i (xi − x̄)2 = 5540, ∑ i (yi−ȳ)2 = 0.756, ∑ i (xi−x̄)(yi−ȳ) = −41.9 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 30 40 50 60 70 0.5 0.6 0.7 0.8 0.9 1.0 Age D en si ty (a) Find the simple linear regression line. (b) Carry out a hypothesis test that the slope of the regression line is zero versus the alternative that it is negative. Interpret the results of this test in the context of the problem. (c) Use the regression equation to predict the bone densities of a sixteen (16) year old girl and a fifty (50) year old woman. (d) The full widths of the two 95% prediction intervals based on the equation on page 374 of the Course Notes are respectively 0.47 and 0.43 for the 16-year-old girl and 50-year-old woman. Briefly explain why one of these prediction intervals is much more reliable than the other.