Statistics 311 Final Exam Instructions and Problems, Exams of Probability and Statistics

Download Statistics 311 Final Exam Instructions and Problems and more Exams Probability and Statistics in PDF only on Docsity! Name:__________________ Section: Statistics 311 Final Exam 11 December 2007 E:\Stat311\Autumn2007\FinalExamAu07Solutions.doc 1 __________ Please read the following instructions carefully. Do not turn the page and start the exam until you are told to begin. You may have the formula card that comes with the textbook (or a copy of the same). You may also have one sheet (double-sided) of 8.5 x 11 paper with notes (handwritten or typed). You may use a calculator, although be sure to show your work. There are 3 multi-part short answer problems worth a total of 98 points; you get 1 point for your name and 1 point for your quiz section. In addition, there are 5 multiple choice problems and 5 true/false problems. The multiple choice and true/false problems are worth one point each for up to 10 points of extra credit. Point values for each question or part of a question are designated in parentheses at the beginning of the problem. GOOD LUCK! *The data set is from exercise 10.64 in Neil Weiss’ Introductory Statistics, 8th edition, 2008, Pearson Education, Inc. Data are from a paper by Lee and Dunton, Marine Ecology Progress Series, 2000, Vol. 196, pp. 39-48. 2 Problem 1 (6+10+4+6): Researchers of the Marine Science Institute of the University of Texas at Austin noticed a difference in the seagrass beds in Corpus Christi Bay (CCB) versus those in Lower Laguna Madre (LLM). Researchers know that ammonium is essential to seagrass growth. As such, they decided to compare sediment (pore water) ammonium concentrations (in Mµ ) at the two locations. Summary statistics based on collected samples are given below*. CCB CCB CCB LLM LLM LLM 115 1 79 4 51 24 3 10 5 19 x . s . n x . s . n = = = = = = a) Write down the null and alternative hypotheses to test whether, on average, ammonium concentrations at CCB are greater than those at LLM. Be sure to appropriate label your parameters. 0 1: or = 0 versus : or 0CCB LLM CCB LLM CCB LLM CCB LLMH Hµ µ µ µ µ µ µ µ= − > − > b) Construct the appropriate test statistic for your test in part a) using an un-pooled procedure. ( ) ( ) 2 2 0 115 1 24 3 0 90 8 90 8 90 8 7 98 11 3766304 36 110 25 123 615 5 803 129 418 51 19 CCB LLM obs CCB LLM CCB LLM x x . . . . .t . .. . . . .s s n n − − − − = = = = = = + ++ ( ).05 1 ,18 1.734t = c) Using a 5% significance level, what is the critical value for this test? d) Using your test statistic and critical value, state the conclusion from your hypothesis test in relation to the original question. Since 7.98 > 1.734, reject the null and conclude that sample evidence supports that the mean ammonium concentration at CCB is greater than that the mean concentration at LLM. 5 Problem 3 (15+5+4 points): In a survey of 190 college students, 134 students said that they believe there is extraterrestrial life. a) Construct a 99% confidence interval for the proportion of all college students that believe there is extraterrestrial life. ( ) ( )( ) ( )( ) ( )( ) ( ) 2 .705 .295ˆ ˆ134ˆ ˆ0.705; 0.705 2.575 190 190 0.705 2.575 .0331 0.705 2.575 .085 0.62,0.79 pqp p z nα= = ± ⇒ ± ⇒ ± ⇒ ± ⇒ b) Does the confidence interval support the notion that the proportion of college students that believe there is extraterrestrial life exceeds 68%? Briefly explain. No, since the lower bound of the CI includes 0.62 which is less than 68%. c) If you had constructed a 90% interval instead of, would you expect this interval to be wider or narrower than the 99% interval? Briefly explain. The 90% CI would be narrower since we are willing to be less confident. 2 zα would be smaller, making E smaller. EXTRA CREDIT EC1 (1 point each): Answer the following multiple choice questions by circling the letter with the correct response: a) Which of the following quantities does NOT affect the width of a confidence interval for a population proportion? A. The confidence level. B. The sample proportion. C. The sample size. D. The population size. b) When constructing a confidence interval for a population mean µ , for a sample of n = 29 observations and σ unknown, the correct formula to calculate the margin of error is A. 2E z nα σ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ B. ( )2 ,28 sE t nα ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ C. None of the above. 6 e) The likelihood that a statistic would be as extreme or more extreme than what is observed based on the sample data is called a A. statistically significant result. B. test statistic. C. significance level. D. P-value. f) The smaller the P-value, the A. stronger the evidence against the alternative hypothesis. B. stronger the evidence for the null hypothesis. C. stronger the evidence against the null hypothesis. D. None of the above. g) Which statement is NOT true about hypothesis tests? A. Hypothesis tests are only valid when the sample is representative of the population for the question of interest. B. Hypotheses are statements about the population represented by the samples. C. Hypotheses are statements about the sample (or samples) from the population. D. Conclusions are statements about the population represented by the samples. EC 2 (1 point each). Indicate whether each of the following statements is True or False. False A confidence interval for the average value of y at a given value 0x x= , formed as part of a linear regression analysis, will always be wider than the corresponding prediction interval estimate for a new individual value of y at the same value 0x x= . True One assumption for simple linear regression is that the epsilons are normally distributed with a mean of zero. True Power is equal to 1 β− . False The level of significance associated with a significance test is the probability of committing a Type II error. True A statistical hypothesis test is always conducted under the assumption that the null hypothesis is true.