Reference materials;

ducks.mwx

In this EMA, you will analyse some data on the survival times of ducks described below. Some female ducks, taken from two locations in New J ersey, were captured and tted with radios over a period of about one month= starting in midNovember one particular year. Their progress was then closely monitored until late February the following year, when the experiment was terminated. Eighteen birds died eight from natural causes and ten due to hunters. The hunting season ended in midFebruary. The survival time is either the time until death or the time up to the end of the period of observation. [Data source: Pollock, K.H., W'interstein, SR. and Conroy, Md. (1989] 'Estimation and analysis of survival distributions for radiotagged animals', Biometrics, vol. 45, pp. 99109.) Question 1 3 marks Data on 50 of the ducks are given in the le ducks.mwx. In this le there are the following variables. 0 duck: a serial number that identies each duck. o survival: survival time [in days}. 0 death: '0' the duck was still alive at the end of the observation period? '1' the duck had died by the end of the observation period. 0 age: '05 the duck was a hatchyear duck (i.e. in its rst year of life}= '1' the duck was an adult (i.e. at least one year old). 0 weight: the weight [in grams} of the duck at the time of capture. 0 length: the winglength [in millimetres) of the duck at the time of capture. One duck was lighter than all the other ducks. By looking at the data given in the le ducks.n1wx, answer the following. (a) lill-That is the serial number of this duck? (b) lill-That was the winglength of this duck? 'Was it also the duck with the shortest winglength? Question 2 5 marks (a) Using Minitab, produce a stemplot of the weights of the ducks. Include a copy of this stemplot in your answer. (When producing the stemplot, you should leave the Increment eld blank and the Trim outliers option unselected, and make sure that the stemplot includes all the ducks.) (b) Use your stemplot to describe the shape of the distribution of weights of the ducks. Justify your answer. Question 3 18 marks Using the data given in ducks.mwx, a researcher wishes to construct a model to predict the winglength of a duck based on its weight. (a) Briey explain why the researcher should treat weight as an explanatory variable and winglength as a dependent variable. (b) Obtain a scatterplot of winglength against weight. Include this scatterplot in your answer. Interpret this scatterplot, giving reasons for your interpretation. (c) 1'What is the equation of the least squares regression line tted to these data? INith reference to a suitable residual plot [which you should include in your answer)= briey discuss whether or not the least squares regression line ts the data well. ((1) Using the line tted in part (c), predict the winglength of a duck that weighs 1200 g. Also state the 95% prediction interval for the winglength of a duck that weighs 1200 g. Give both of your answers rounded to the nearest whole number. Interpret the prediction interval. Question 4 13 marks A contingency table of the age of the ducks and whether or not their deaths had been observed is given below. Table 1 Age Hatchyear duck Adult duck Total Death not observed 19 13 32 Death observed 12 6 18 Total 31 19 50 Note that this table is given in the le duck death.rnwx. (a) Explain carefully why this table is correctly described as a contingency table. (b) 1'What is the probability that a randomly selected duck from this group was observed to have died? Give your answer to three decimal places. (c) 1What is the probability that a randomly selected duck from this group was observed to have died given that it was a hatchyear duck? Give your answer to three decimal places. ((1) Suppose that a researcher is interested in the following question: Does the age of a. duck aect whether or not the death of the duck was observed? (i) W'rite down suitable null and alternative hypotheses. (ii) Using the X2 test for contingency tables, test the hypotheses you wrote down in part (d)[i). Make sure to include in your answer: a a table of the expected values [with the expected values given to two decimal places) the value of the test statistic the degrees of freedom the p-value, or the values of CV5 and CV1 your conclusion from the test. Question 5 11 marks (a) (b) ((1) Using Minitab, or otherwise, produce a diagram containing boxplots of the weights for the hatch-year ducks and the weights of the adults. (These weights are given in separate columns, hatch-year and adult= of the le duck weightsnlwx.) Ensure that the boxplots are horizontal, have the same scale and are drawn on one diagram. Prepare these boxplots ready for inclusion in a report by ensuring that the title and the horizontal axis label are clear and informative. Include the finished boxplots in your answer. Complete the following table of summary statistics for the weights of ducks. (All the values should be rounded to the nearest whole number.) Table 2 Hatchyear ducks Adult ducks Numb er Mean Median Standard deviation Interquartile range Range Using your answer to part. (a), compare the skewness of the distribution of weights of hatchyear ducks to the skewness of the distribution of weights of adult ducks. Using your answer to part. (b), calculate by hand the estimated standard error for the mean weight of hatchyear ducks. Show your working and give your answer to one decimal place. Question 6 14 marks In Question 55 you compared the distributions of weights of hatchyear and adult ducks. In this question you will consider one aspect of the distribution the mean weight. Suppose now that interest focuses on whether the population mean weight of ducks is the same for hatchyear and adult ducks. (a) (b) lNrite down suitable null and alternative hypotheses= stating clearly the meaning of any symbols that you use. In part (d)= you will use a two~sample ttest to test. the hypotheses that you wrote down in part (a), using the data in duck weightsrnwx. Explain why neither the twosample ztest nor the paired ttest would be suitable for these data. For the data in duck vsreightsnnvvx= is the assumption of a common population variance reasonable? Justify your opinion. State any other assumptions that are made about the data. (d) Use the twosample ttest to test the hypotheses that you wrote down in part (a). [Whatever answer you gave in part (c), you should assume that the assumption of a common population variance is indeed reasonable, along with any other necessary assumptions.) In your answer, make sure to include: o the value of the pooled estimate of the common standard deviation 0 the value of the test statistic o the degrees of freedom 0 the pvalue, or the values of CV5 and CV1 c any conclusions that can be drawn from the results of this test. [6] Question 7 3 marks Suppose that the original researchers were interested in taking more detailed measurements of the ducks that were initially captured, but could aord to do this with only 5 of the 50 ducks. Using simple random sampling, select a sample that the researchers could have used. To help you to select this sample, you may use either Minitab or the random number table given in the appendix to Unit 4. In your answer, you should outline the method that you used to obtain the sample, and detail which random numbers you generate [including any that you discard). If you use Minitab, then you should make sure that the seed of the random number generator is initially set to take the value 1058. If you use the random number table, then you should start at row 8. [3] Question 8 3 marks A different researcher would like to test a new treatment for seabirds that get contaminated after an oil spill. The next time there is an oil spill, she proposes taking pairs of birds and using this new treatment on one bird in each pair and applying a standard treatment to the other. The outcome is then whether or not each bird is fit enough to be released back into the wild. (a) Explain why the data from this trial would be regarded as objective data. [1] (b) Explain why the data from this trial relating to each bird would be regarded as categorical data. [1] (c) How should the pairs of birds be selected to ensure that this is a matchedpairs trial? [1] C1 C2 hatch-year adult 1140 1160 2 1160 1260 3 1120 1080 4 1070 1140 5 940 1200 6 1240 1100 1120 1420 8 1010 1120 9 1040 1110 10 1200 1340 11 1280 1250 12 1250 1050 13 1090 1320 14 1040 1260 15 1130 1110 16 1180 1280 17 1070 1270 18 1270 1370 19 1080 1220 20 1150 21 1030 22 1160 23 1180 24 1050 25 1050 26 1160 27 1150 28 1220 29 1 140 30 1140 31 1120hatch-year adult 7 death not observed 19 13 2 death observed 12 6Appendix: random number table This table contains 3000 random digits (i.e. throws of a ten-sided die labelled 0, 1, . .., 9). 1 98 06 77 46 16 63 998081 82 1548 96 1256 51 646021 124160 046393 45 25 52 75 50 35 41 25 66 21 51 66 11 34 33 18 36 41 33 18 70 52 64 76 41 17 07 54 01 29 86 41 93 16 55 54 40 68 58 71 24 92 06 94 84 48 92 96 32 29 00 60 53 93 46 82 67 64 48 91 74 85 94 40 51 30 93 08 78 32 89 76 61 03 01 20 94 36 39 87 52 27 23 54 82 64 44 58 45 94 30 39 86 19 64 84 35 30 19 5 50 76 11 36 1384 3293 72 29 04 41 25 43 89 55 61 46 40 89 21 47 20 85 91 90 56 67 40 31 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