# Math 130 Final Exam Spring 2014 \ NAME: . You must show all work, calculations, formulas used to receive any credit. NO WORK =NO CREDIT.

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1 Math 130 Final Exam Spring 2014 \ NAME:. You must show all work, calculations, formulas used to receive any credit. NO WORK =NO CREDIT. Round the final answers to 3 decimal places. Good luck! Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Total out of 100.

2 Question 1. Determine whether a hypothesis test or confidence interval from the 5 main scenarios, or some other analysis (regression, ANOVA, chi-square) is appropriate for each research question. If you select other, you need to specify the other procedure. Topics were chosen from most recent issue of the Journal of Agricultural, Biological, and Environmental Statistics. a. A study compared the toxicity on flies of four different types of selenium. The number of dead flies was counted and the selenium type (type 1, 2, 3, or 4) was recorded for each observation. The researchers want to know if the toxicities differ between selenium types. Hypothesis Test Confidence Interval Other: b. A study conducted in Australia measured crop yields for wheat and lupin on many different fields and researchers want to estimate the difference in mean crop yields for the two crops. Hypothesis Test Confidence Interval Other: c. A study wanted to examine the relationship between number of Japanese beetle grubs and percentage of organic matter in the soil for locations on a golf course in New York. The researchers want to know if higher grub numbers are associated with lower percentages of organic matter. Hypothesis Test Confidence Interval Other: Question 2 A quick perusal of the Audobon Society Field Guide to mushrooms reveals that mushrooms can be classified by a variety of characteristics.one possible habitat for mushrooms is grasses. Suppose you take a random sample of 100 mushroom entries from the field guide and you find that 36 of them have grass habitats. Now assume that you had previously heard that 40 percent of mushrooms live in grasses. a. What is the sample proportion of mushrooms with grass habitats?

3 b. State appropriate hypotheses to test if your sample is evidence that fewer than 40% of mushrooms live in grasses. Use a significance level of.01, and be sure to discuss any possible issues with conditions necessary for carrying out the test. State hypotheses: Assumptions: Test statistic: p-value computations: p-value interpretation: What is the distribution of the test statistic assuming the null hypothesis is true? Conclusion:

4 c) Suppose you had decided to do this test by making a confidence interval. What confidence level for a CI is consistent with performing your test at a.01 significance level? What is your decision using this confidence interval? Question 3 We continue our investigation of the mushroom data with a larger random sample and a new variable called "growth pattern". The possible values for this variable are abundant, clustered, numerous, scattered, several, and solitary. A random sample of mushrooms yields the following summary information: Growth Abundant/Clustered/Numerous Scattered Several Solitary Total Pattern Obs.Count = a. Suppose you believe that the growth pattern several occurs for 50% of mushrooms, scattered accounts for 12.5%, solitary accounts for 25%, and when abundant, clustered, and numerous are considered together, they account for 12.5% of mushrooms. What test should you use to test your belief? b. Set up appropriate hypotheses to test the belief.

5 c. Comment on the conditions that need to be satisfied for your inference procedure. You should add some helpful numbers to the table above. d. Complete the mechanics of your test procedure by computing the test statistic and p-value. e. What distribution did you use to find the p-value? f. Interpret your p-value in context. g. What is your decision at a.01 significance level?

6 Question 4 Returning one last time to the mushroom data set, we consider results of one much larger random sample of mushrooms. The variables under consideration are bruises (yes or no) and number of rings (zero, one, or two). The data are summarized in the following table: Bruises/Ring Number Zero One Two Total Yes 0 ( ) 3080 ( ) 296 ( ) No 36 ( ) 4408 ( ) 304 ( ) Total a. Suppose you want to know if there is an association between bruise status and ring number. What is the appropriate analysis to run? b. Are the conditions met for your analysis selected in a.? Support your answer by filling in expected counts in the table above. c. Set up appropriate hypotheses. Complete the mechanics of your test procedure by computing the test statistic and p-value.

7 d. What distribution did you use to find the p-value? e. Interpret your p-value in context. f. What is your decision at a.05 significance level? g. What type of error could you make in your hypothesis test? Type 1 Type 2 None

8 Question 5 A biologist studying lizards, specifically Cophosaurus texanus, recorded the weight (mass) in grams, snout-vent length (SVL) and hind limb span (HLS) of a random sample of 25 such lizards. The biologist wants to study the relationship between variables, looking to see if SVL can be used to predict weight (mass) accurately. A basic scatterplot shows the data at right. a. Based on the scatterplot, how would you describe the relationship between SVL and mass? A student working in the biologist s lab runs a regression analysis on the data and produces the following partial Rcmdr output: Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-10 *** SVL e-15 *** --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 23 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 23 DF, p-value: 3.836e-15 b. What is the value of the correlation coefficient? Interpret the correlation coefficient. c. Interpret the R-squared value. How well you think this model fits the data?

9 d. What is the equation of the least squares regression line e Check the assumptions. f. Now, we want to assess whether or not SVL can be used to predict mass (weight).what hypotheses correspond to determining if SVL is a significant predictor of mass( weight)? Null hypothesis: Alternative Hypothesis : Test Statistic: Distribution of Test Stat: p-value : Conclusion:

10 g. Obtain a 99% confidence interval for the population slope. (You do not need to list assumptions.) Interpret your interval Can you conclude the population slope is less than 1? Explain. h. Obtain predictions for mass based on SVLs of 70 and 100, if appropriate. If inappropriate, explain why. i. Compute a 99% prediction interval for an individual response when SLVs=70. (Assume that the average SLVs= 68) Interpret your interval.

11 j. Compute a 99% confidence interval for the mean response when SLVs=70. Interpret your interval Question 6 A 2008 study in Ecology examined the effect of parasites on the nutritional quality of the host, when considered as a food resource for predators in the context of Daphnia (small, planktonic crustaceans) with a parasitic infection by Chytridiomycete. The crustaceans are a food resource for other organisms. The researchers studied the levels of various fatty acids in gravid (pregnant) Daphnia, uninfected Daphnia, and infected Daphnia to see if the infected ones were adversely affected via an ANOVA analysis. We will focus on the levels of HUFA - highly unsaturated fatty acids. There were 15 Daphnia observed from each group. Data was generated to be consistent with summary statistics in the article. a. The ANOVA performed was balanced unbalanced. b. What condition can the boxplots be used to check that is NOT similar to a condition required for a two sample t-test? Does the condition appear to check out? Explain.

12 c. A partial ANOVA table was provided as: Complete the table. DF SS MS F p-value Daphnia Type e-11 Residuals d. What is your best estimate of the common population variance? e. What is the distribution of the test statistic assuming the ANOVA null hypothesis is true? f. Interpret the ANOVA p-value in context. g. What is your decision at a.01 significance level? h. The following output was also generated. If appropriate to use, what does it tell you about the HUFA levels in the 3 groups of Daphnia? If not appropriate to use, explain why not. (1=Gravid, 2=Infected, 3=Uninfected) Estimate lwr upr

13 Question 7 Lizard measurements of mass and snout-vent length (SVL) for 2 genera Cnemidophorus and Sceloporus - were collected in 1997 and The primary researcher wants to know whether or not Cnemidophorus has a smaller SVL than Scelophorus, on average. Observations were collected for a random sample of 20 Cnemidophorus and 40 Sceloporus lizards. a. Explain in one sentence why a paired t-test is not appropriate for this data set and research question. b. Set up appropriate hypotheses and parameter definitions to address the researcher s question. Null: Alternative: Where c. Assuming the conditions checked out, the following Rcmdr output was obtained. The subtraction order was Cneidophorus Sceloporus. Welch Two Sample t-test data: svl by genera t = , df = , p-value = alternative hypothesis: true difference in means is not equal to 0 mean in group Cnemidophorus mean in group Sceloporus Test: p-value : Provide an appropriate conclusion at α=0.05 significance level.

14 Question 8 A bottle machine can be regulated so that the amount of fill dispensed by the machine per bottle, Y, is distributed with mean µ ounces and standard deviation of σ ounces. A large sample of n filled bottles is randomly selected from the output of the machine on a given day (all bottled with the same machine setting), and the ounces of fill, Y 1,Y 2,.Y n are measured for each bottle. a) Describe the sampling distribution of. b) Suppose the population mean µ is unknown, but the population standard deviation = 5 ounces and n=100. Find the probability that will be within 2 standard deviations from the true population mean µ.

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