Stat 111 - Introduction to Statistics Final Exam

Name: Stat 111 - Introduction to Statistics Final Exam Tuesday, May 9th 9 am -12 noon Instructions: 1. Show all work. You may receive partial credit for partially completed problems. 2. You may use calculators and three one-sided sheets of reference notes (or equivalent). You may not use any other references or any texts. You may NOT use your cell phone as your calculator. 3. You may not discuss the exam with anyone but Prof. Wagaman. Uphold the Honor Code. Cell phones and mobile devices must be turned off and put away in your backpacks. 4. Suggestion: Skim/read all questions before beginning and complete the ones you know best first. Point values per problem are displayed below. More space than needed is provided. 5. Carry calculations out to four decimals, unless otherwise specified. I.E. 3/10 would be 0.3 but 2/3 would be 0.6667. 6. If you cannot solve an earlier part of a problem, and a later part depends on that part, use a reasonable value and state you are ASSUMING that is the right value to use. This allows you to still get credit for later parts if you get stuck on something early on. 7. For stating and checking assumptions – BE SPECIFIC. Saying "nearly normal condition" by itself does not tell me what you think must be nearly normal. A check mark does not show me you know how to check the conditions. Demonstrate that you know what to check. 8. Interpretations/conclusions/any written statement must ALWAYS be in context of the problem. 9. Initial in the box to acknowledge that you have read and understand the instructions, especially in regard to #8. 10. Good luck! Problem 1 2 3 4 5 6 7 8 Total 16 8 10 18 12 11 16 9 100 Points Earned Possible Points 1 1. The fog index is a measure of reading difficulty based on the average number of words per sentence and the percentage of words with three or more syllables. High values of the fog index are associated with difficult reading levels. Independent random samples of 6 advertisements were taken from Scientific American, Fortune, and the New Yorker magazines. We want to see if there is a difference in average fog level between any of the magazines. Use the output below to help you address the questions that follow. Analysis of Variance Table Response: fog Df Sum Sq Mean Sq F value Pr(>F) magazine 2 48.529 24.2644 ?????? 0.007234 Residuals 15 52.217 3.4812 a. (2 points) One of the conditions for the appropriate test procedure to address the research question is that the ________ (#) populations of magazines must have distributions of ________ ____________ with equal ________________________. Our best estimate of that common value is ____________ (#), based on our output. b. (2 points) Compute the missing value of the test statistic. c. (3 points) Sketch the distribution used to compute the p-value. Clearly label the distribution, specifying all necessary parameters, and your test statistic. Shade in the area indicating the p-value. 2 1.d. (4 points) State the value of and interpret the p-value obtained. Hint: Use your picture from c. to help. e. (2 points) Using a 0.01 significance level, we have sufficient evidence to conclude that… Some additional R output was generated. with(fog, pairwise.t.test(fog, magazine, p.adj='bonf')) Pairwise comparisons using t tests with pooled SD data: fog and magazine Fortune NewYorker NewYorker 0.022 ScientificAmerican 1.000 0.013 P value adjustment method: bonferroni f. (1 point) This output is an example of ______________________ _________________________ output. g. (2 points) If appropriate, summarize any differences in average fog index identified via the output. If not appropriate, explain why not. 3 2. This semester, we worked with the Kenyon Pines data set in many example R problems. Let’s take a last look at a variable in this data set. Grow96 is the recorded leader growth for the white pine trees in 1996, measured in cm. Some R output for this variable is provided. Use the output to address the questions that follow. favstats(~Grow96,data=Pines) min Q1 median Q3 max mean sd n missing 0 64 78 90 126 74.17037 21.21238 864 136 a. (2 points) Grow96 is a (circle all that apply)… discrete continuous categorical quantitative variable. b. (2 points) Describe the distribution of Grow96. (Remember: 3 items!) c. (2 points) Name a feature of the distribution of Grow96 that is visible in the histogram, but NOT in the boxplot. d. (2 points) Verify that the boxplot of Grow96 should have outliers, using a calculation as support. 4 3. A data set on fish in a particular estuary contains a number of variables collected by biologists. Variables included are the fish species, weight, width, height, and several lengths. For each research question below, determine the appropriate analysis to pursue from among our inferential procedures this semester. If you choose hypothesis test or CI, you must specify the parameter of interest. Similarly, if you choose a chi-square test, you must specify which test. a. (2 points) Research question: Biologists want to be able to predict fish weight using length (L3), assuming that there is an appropriate relationship between the two variables. b. (2 points) Research question: The biologists want to know if the fish weigh less on average now, than a few years ago, when the average weight was 3 pounds, due to changes in the environment. c. (2 points) Research question: Biologists have a breakdown of the fish species (% composition) from three years ago before some flooding. They want to see if the current distribution matches the preflooding distribution. d. (2 points) Research question: The biologists want to know if more than 35 percent of the fish have a width that is “large” (where “large” has been set by threshold that varies by species). e. (2 points) Research question: The biologists are interested in how long the tails of the fish are. Luckily, one length variable runs from the nose to the end of the tail, and a second length variable runs from the nose to the start of the tail on the fish. 5 4. An article “…for Forecasting Critical Fish Influxes at Power Station Intakes" (J. Applied Ecology. (1983): 33-42) contains data on the number of fish caught at power station intakes as well as the water temperature, minimum tide height, number of pumps running, pump speed, and wind-range of direction (degrees). You may assume it is a representative sample. We want to predict the number of fish caught so we can deduce what weather conditions are most dangerous in terms of clogging up the intakes with fish. Use the output provided to answer the questions below. After some trial and error, the following model was fit: mod=lm(fish~watertemp*pumps+mintide+pumpspeed+windrange,data=fish) summary(mod) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -120.34609 107.92286 -1.115 0.2795 watertemp 18.38152 9.60268 1.914 0.0716 . pumps 44.85500 32.17559 1.394 0.1803 mintide 7.20858 4.94398 1.458 0.1621 pumpspeed 1.48920 0.58993 2.524 0.0212 * windrange 0.09472 0.05302 1.787 0.0908 . watertemp:pumps -6.77573 3.04516 -2.225 0.0391 * Residual standard error: 10.23 on 18 degrees of freedom Multiple R-squared: 0.482, Adjusted R-squared: 0.3093 F-statistic: 2.792 on 6 and 18 DF, p-value: 0.0424 a. (2 points) This model is able to explain _______________ % of the observed variability in __________ using its linear relationship with the other variables. b. (3 points) Interpret the slope coefficient of pumpspeed in context. favstats(~fish,data=fish) min Q1 median Q3 max mean sd n missing 1 7 11 14 53 13.8 12.30515 25 0 c. (3 points) How well does the model fit? Explain, using at least 2 numeric values as support for your response. 6 4.d. (3 points) Is the overall model useful for predicting the number of fish caught at the intakes? Provide an appropriate test statistic, p-value, and conclusion at a 0.10 significance level. Assume all necessary conditions for inference hold. Test statistic: ______ = _______________ p-value: Conclusion: e. (4 points) The last term in the model output is watertemp:pumps. This is an interaction term between watertemp and pumps. While we did not discuss these types of terms in class, they can be useful to include in models, where the effect of one variable is moderated by the value of another variable. In OUR model, does the interaction term appear to be a significant predictor? Provide hypotheses, an appropriate test statistic, p-value, and conclusion at a 0.10 significance level. Assume all necessary conditions for inference hold. Null: Alternative: Test statistic: ______ = _______________ p-value: Conclusion: f. (3 points) The plot at right is a _______________ plot of the residuals which is used to check one regression assumption in particular. The condition is that the _____________ ______________ are normally distributed. We do not have those values, so we check the condition using the _____________________ instead. Here, it appears that the condition is met not met because… 7 Hammock\DBH Group Deering Matheson Total 2-4 Tiny 170 165 335 4-6 Small 79 88 167 5. Hurricanes can have substantial ecological consequences depending on their severity. After Hurricane Andrew in 1992, some ecologists decided to study how well exotic plants recovered compared to native plants in different hardwood hammocks. For part of their analysis, they simply counted the number of plants in certain diameter at breast height (DBH) groups (in centimeters) eight months after the hurricane. Here, we are focused on the smallest plants – those in either the tiny or small DBH categories. You may assume the randomization and independence conditions are satisfied. a. (1 point) Provide a statistical estimate (one #) of the difference in fractions of tiny plants out of the smallest plants (tiny and small combined) between these two hammocks. b. (2 points) Provide the numeric value of the standard error of the estimate you computed in part a. c. (3 points) To help improve the estimate, provide an appropriate 90% confidence interval for the difference in fraction of tiny plants out of small plants between these two hammocks. Be sure to check any remaining conditions. d. (3 points) Interpret the confidence level in context. e. (3 points) At a significance level of 0.10, our confidence interval does does not provide sufficient evidence to allow us to conclude that there is a difference in the fraction of tiny plants out of small plants between these two hammocks because …. 8 Total 249 253 502 6. A student wants to study college student use of pens versus pencils, and designs a survey to ask students whether they prefer pen or pencil. The student believes that pens should be used 70% of the time (because, really, how often do you need to erase things anyway?). From a random sample of 35 students, he learns that 21 students prefer pen (and the other 14 prefer pencil). a. (2 points) Suppose X is defined as the random variable which counts the number of students out of the 35 who prefer pens, according to the student’s belief. Then, you would want to model X using a binomial uniform normal model with parameters _________________________. (Be sure to fully specify all necessary parameters for the model). Now suppose the student is more comfortable working with fractions, so he swaps to working with X/35, the sample proportion. Address the questions below for this setting. b. (1 point) The _________________distribution of a statistic is the distribution you get by considering the values of the statistic for all possible samples of a set size from the population. c. (3 points) Obtain the relevant distribution (from b.) for the sample proportion of students who prefer pen assuming the students' belief is correct. Be sure to specify all necessary parameters, and check any necessary conditions beyond the randomization and independence condition. d. (3 points) How unusual is the sample proportion of 21/35=0.60 assuming the student’s belief is correct? Explain. Use the R output to provide numeric support for your response. pnorm(0.70,0.60,sqrt(0.60*0.40/35)) 0.8864023 pnorm(0.60,0.70,sqrt(0.70*0.30/35)) 0.0983528 pnorm(0.60) 0.7257469 e. (2 points) Provide the upper 16% cutoff for the sample proportion of students (out of 35) who would prefer pen assuming the student’s belief is true. (Note: suggest drawing a picture to help; no R code needed). 9 Mold Spore Level Death Non-death 7. A 1995 article from the Journal of Allergy and 0-99 (Low) 15 280 Clinical Immunology is titled: "Effect of 100-499 (Moderate) 11 235 environmental molds on risk of death from asthma during the pollen season." As part of 500-999 (High) 9 163 their findings, a series of tables is presented. We will focus on Table 3, which has the number of 1000+ (Very High) 32 287 death days (at least one death on the day from Total 67 965 an asthma-related issue) from 5-34 year olds in Chicago from the pollen season of 1985-1989 broken down by mold spore level. Use the table to help address the questions that follow. Total 295 246 172 319 1032 a. (1 point) If you randomly selected a day from the study, what is the probability it would have a moderate mold spore level? b. (2 points) If you randomly selected a death day from the study, what is the probability it would have a very high mold level? c. (2 points) (General) What are the four main steps in a hypothesis test procedure? List in order. d. (1 point) The authors want to know if there is a relationship between mold spore level and death day status. What test procedure would you recommend? (Be specific). e. (2 points) What are the null and alternative hypotheses for your chosen test procedure? 10 The data was entered into R and the following output obtained. Use the output to address the remaining questions. xchisq.test(tally(moldlevel~deathday)) Pearson's Chi-squared test X-squared = 9.6485, df = 3, p-value = 0.0218 9 163 ( 11.17) (160.83) [0.420] [0.029] 15 280 ( 19.15) (275.85) [0.900] [0.062] 11 235 ( 15.97) (230.03) [1.547] [0.107] 32 287 ( 20.71) (298.29) [6.154] [0.427] key: observed (expected) [contribution to X-squared] 7.f. (2 points) State the condition related to expected counts in this setting, and state whether or not the condition checks out, with support for your response. g. (3 points) Sketch and label the distribution used to find the p-value, showing the test statistic, and shading in the p-value. h. (3 points) Circle your decision at a .05 significance level and what type of mistake you could have made, and complete the sentence. Decision: Reject Mistake: Type I Not reject the null hypothesis Type II error If you wanted to reduce the probability of your selected type of error, one option would be to Increase decrease the significance level used. 11 8. Does having an advanced degree really lead to higher incomes? To help address the question, we consider the following data: the square root of incomes in U.S. dollars for 1,020 working Americans who had 12 years of education and 406 working Americans who had 16 years of education, in 2005. (Eg. 200 units for sqrt income = 40000 dollars of income). You may assume this is a representative sample of American adults. a. (2 points) Based on the boxplots, would you expect the mean for each group (either 12 years or 16 years) to be larger than the median? Explain. b. (3 points) We want to test whether there is a significant difference in sqrt(Income) between the two education groups. One of the conditions for the test is related to normality. State the relevant condition, and whether or not the condition checks out, with support for your response. Some additional R output was obtained. t.test(sqrt(Income2005)~Educ) Welch Two Sample t-test data: sqrt(Income2005) by Educ t = -11.558, df = 547.27, p-value < 2.2e-16 alternative hypothesis: true difference in means is not equal to 0 mean in group 12 mean in group 16 179.9095 243.7655 c. (4 points) A journalist sees the R output and writes the following headline: “Get More Schooling! Experiment shows it is practically impossible to have equal incomes without equal education.” Critique the journalist’s choice of headline from a statistical viewpoint, and include at least one of the following terms in your response: experiment, observational study, causation. 12