HW12_Q_Spr2009 - STAT 350 – Spring 2009 Homework #12...

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Unformatted text preview: STAT 350 – Spring 2009 Homework #12 covers through Lecture S 1. An ecologist is studying the trade-offs individuals make between investment in survival and growth and the investment in reproduction. In particular she is studying this relationship in a highly variable species of prairie grasses. She has sampled 25 individual (randomly selected) plants across the range of this species. For each plant, she has measured seed mass and root volume. Let xi denote the root volume (in mL) of the ith plant and yi denote the seed mass (in mg) 25 of the ith plant. She finds 25 ∑ x = 926 , ∑ y i =1 i i =1 i 25 = 1,928, ∑x i =1 2 i 25 = 51,860, ∑y i =1 2 i = 153,997, and 25 ∑ ( x y ) = 63,677. i =1 i i The root volumes ranged from 8.2 to 102.3 mL and the seed masses ranged from 44.7 to 102.5 mg. Warning: do not round-off too much in intermediate calculations! a. b. c. d. e. f. g. h. i. j. k. l. m. n. o. p. q. r. Calculate SSxy. Calculate SSxx. Calculate SSyy. Calculate the standard deviation of root volume Calculate the standard deviation of seed mass Give a 95% CI for the true mean seed mass of individuals from this species. The biologist finds another individual of this species, but during a time of year when the plant is not producing seeds, give a 95% prediction interval for the seed mass that would be produced by this individual. Find the Pearson Correlation between the root volume and seed mass of this species. Find the least squares regression equation to predict seed mass from root volume for this species (give a and b to at least 5 decimal places). Create the ANOVA table for the regression analysis (use Excel to obtain the p-value for your table). What percent of the total variation in seed mass is explained by the linear relationship between root volume and seed mass? Find the standard deviation of the least squares line (se) Obtain a 95% CI for β (the true average increase in seed mass with a 1-mL increase in root volume). Based on your answers to parts (j) through (m), what can you conclude about the relationship between root volume and seed mass in this species? Be sure to use proper statistical language and support your answer appropriately! The biologist finds another individual of this species, but during a time of year when the plant is not producing seeds. However, she can measure the root volume and finds that it is 40 mL. Give a 95% prediction interval for the seed mass that would be produced by this individual. The biologist finds another individual of this species, during a time of year when the plant is not producing seeds. She measures the root volume and finds that it is 100 mL. Give a 95% prediction interval for the seed mass that would be produced by this individual. Compare the 3 prediction intervals obtained from parts (g), (o), and (p). Which is the narrowest? Which is the widest? Explain why this is. The 8th individual in the sample had root volume = 23.2 mL and seed mass = 74.5 mg. Give the residual for this observation. Homework #12 Page 1 2. For each of the following, determine whether the correlation is "statistically significant" (that is, test the null hypothesis ρ = 0 against the alternative that ρ ≠ 0). Be sure to give the value of the test statistic as well as the p-value. a. r2= 0.01, n = 400 b. r = 0.90, n = 4 3. Jason is a sociology major. For his senior thesis, Jason randomly selected a number of residents from his hometown to survey. He asked each subject a range of demographic questions. Among the questions he asked were: "How many years of schooling have you had?" and "What is your annual income?" Limiting his sample to just those 30 subjects who were no longer in school (that is, who had completed their schooling), the number of years of schooling ranged from 9 to 22 years (mean 15.4 years) and the annual incomes ranged from $28,984 to $61,267. Using these 30 subjects, he conducted a regression analysis to explore whether the amount of schooling affects income. The SAS output from this analysis is given below. Use only the SAS output below and the appendix tables from your textbook to answer the following questions. The REG Procedure Model: MODEL1 Dependent Variable: income Number of Observations Read Number of Observations Used 30 30 Analysis of Variance Source DF Sum of Squares Mean Square Model Error Corrected Total 1 28 29 297006095 1179094138 1476100233 297006095 42110505 Root MSE Dependent Mean Coeff Var 6489.26074 44790 14.48804 R-Square Adj R-Sq F Value Pr > F 7.05 0.0129 0.2012 0.1727 Parameter Estimates Variable Intercept years_school DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 30207 946.97283 5617.60177 356.57432 5.38 2.66 <.0001 0.0129 Questions on next page… Homework #12 Page 2 3 (continued) a. What percent of the variation in incomes is explained by the linear relationship between income and schooling? b. What is the correlation between income and years of schooling? c. Based on the above analysis, what is the income you would expect for an individual from this town who has had 17 years of schooling? d. The 5th subject in this analysis had 17 years of schooling and has an annual income of $41,019. What is the value of the 5th residual? e. How much extra money should an individual in this town expect to earn for every additional year of school he or she has completed? (i) (ii) Give a point estimate. Give a 95% confidence interval f. Jason wants to determine if the relationship between years of schooling and annual income is "statistically significant"? (i) (ii) (iii) (iv) Give the value of the appropriate test statistic Give the degrees of freedom for that test statistic Give the p-value. Based on this, is the relationship "statistically significant"? Just answer "yes" or "no". g. Now assume that Jason wants to test the null hypothesis that years of schooling does not affect annual income (that is, average annual income does not change with an increase in the number of years of schooling) versus the alternative hypothesis that average annual income increases as the number of years of schooling increases. (i) Give the value of the appropriate test statistic (ii) Give the degrees of freedom for that test statistic (iii) Give the p-value. h. According to the Bureau of Labor Statistics, nationwide, average income increased $1750 for each additional year of schooling. Jason wants to compare his town to the national average. He will test the null hypothesis that the trend in his town is the same as the national average against the alternative that the trend in his town is different than the national average. (i) Give the value of the appropriate test statistic (ii) Give the degrees of freedom for that test statistic (iii) Give the p-value. i. j. Give a 95% confidence interval for the true mean income of all residents of this town (who have completed their schooling). Based solely on the preceding analysis, would it be appropriate for Jason to conclude that additional schooling causes increased income? Justify your answer. Homework #12 Page 3 ...
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This note was uploaded on 02/16/2010 for the course MA 350 taught by Professor Sellke during the Spring '10 term at Purdue University-West Lafayette.

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