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University of Waterloo *

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331

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Statistics

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Apr 3, 2024

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pdf

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Uploaded by GeneralStrawHornet26

Tree heights 26 marks A very short, time limited, online quiz was taken by students in a statistics course at the University of Waterloo in 2020. Students were asked two questions and given very little time to answer them. Moreover, they had no idea whatsoever what the questions asked would be about. The information presented in the quiz was as follows: “The coast redwood is perhaps the tallest species of tree growing today. • Do you think the tallest tree of this species alive today is A. less than XXX metres tall? B. more than XXX metres tall? Answer A or B. • Write down your best guess (in metres) of how tall you think the tallest tree might be.” In place of XXX above, about half of the students (randomly selected) had the number 50 appear and the others had the number 100 appear. The value of XXX presented to the students is called the anchor for that question. For the record, and presumably unknown to the students taking the quiz, the tallest coast redwood tree so far found was discovered in 2006. It was named Hyperion after the Titan of Greek mythology of that name (meaning “the high one”) and was measured to be 116.07 metres tall in 2019. The student quiz results are given in the R data file trees.Rda . This may be loaded into R using load() (assuming you have the csv file in a directory/folder given by dataDirectory ) as # Assuming the file is located in the folder/directory given by dataDirectory # For example, a directory/foldr call "data" in the current working directory (".") # dataDirectory <- "./data" load ( file.path (dataDirectory, "trees.Rda" )) # The data are the value of the R data frame called trees head (trees, n = 4 ) ## anchor guess ## 1 100 150 ## 2 100 150 ## 3 100 222 ## 4 100 128 Only the anchor value presented to the student and their guess are recorded (both in metres). The tallest tree is Hyperion # The tallest tree Hyperion <- 116.07 IMPORTANT • In all of your answers, show all the R code you used in your calculations and analyses. • In this assignment, you must write the code using basic R functions like mean() , sd() , var() , sqrt() , length() , pt() , etc. • You may not use functions like t.test() , though these could be used to check your answers. 1

Questions a. First, consider modelling the student guesses according to the mean response model y i = μ + r i for i = 1 , . . . , n where y i is the i th student’s guess of the tallest height. Recall from STAT 231 that to test the hypothesis H 0 : μ = c for some constant c , we form the statistic d = | μ - c | σ/ √ n where μ = y is the arithmetic average (in R mean() ) and σ = ∑ n i =1 r 2 i n - 1 = ∑ n i =1 ( y i - μ ) 2 n - 1 = ∑ n i =1 ( y i - y ) 2 n - 1 is the residual standard deviation (in this case, could use sd() in R ). Large values of d indicate evidence against H 0 and to assess the strength of this evidence, we compute the obseved significance level, or p -value as p = Pr ( | t n - 2 | ≥ d ) = 2 Pr ( t n - 2 ≥ d ) where t n - 2 is a Student’s t random variate on n - 2 degrees of freedom. The smaller is p , the greater is the evidence against H 0 . (See help(pt) in R .) i. (2 marks) Plot a histogram of the guesses (see help(hist) ). Add a “red” vertical dashed line of width 3 at the height obtained by Hyperion. Based only on this display, comment on whether the height of Hyperion might be a reasonable value for μ . Answer # YOUR CODE HERE ii. (1 mark) In R , construct the value of the discrepancy measure d for testing whether the mean guess is the height of Hyperion. Show your code and print the value of d . Answer # YOUR CODE HERE iii. (1 mark) Determine and print the p -value in R for this test. Show your code. Answer # YOUR CODE HERE iv. (1 mark) Based on the above p -value, what do you conclude about the evidence against the hypothesis that the mean of the guesses is the height of Hyperion? Answer b. We now repeat the modelling of part (a), but this time only for guesses from those students who were given the “low” anchor as reference (i.e., anchor == 50 ). i. (2 marks) Select only those students whose anchor == 50 . Using xlim = c(0,400) produce the histogram of the guesses for these students and mark Hyperion with a red dashed line. Comment on whether the Hyperion’s height is a plausible value for μ for these students. Answer 2

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