In [ ]: In [ ]: In [ ]: In [ ]: Question 2. Equipment for measuring concentration of bacteria has standard deviation of the error equalto o = 1.5 + l{]ls bacteria/{ml of fluid). Since error in measurement could be attributed to numerous weakly correlated minor affects, based on a version of central limit theorem with weaker assumptions than independence, we can assume that the error in measurement is (at least sufficiently accurately) normally distributed. Also, assume that the equipment makes unbiased measurements, i.e. the mean of the error in measurement is 0. Denote by 4 the true concentration of the bacteria (unknown to us), and by X = u + error, a single measurement. We saw in class that a linear transformation of a normal random variable (in this case a shift of grror) is also a normal random variable. So, measurement X is a normal random variable. Also, using the basic properties of expected value and variance, we have that E[X] = E[u + error] = u + Elerror] = u as well as var[ X] = var[u + error] = var[error] = oo (You must know these properties of expectation and variance). Thus, we can assume that X ~ N (u; 62). Part (a) Suppose you make 10 measurements of the concentration and get the following values (in millions per ml of fluid), stored as R vector x : x = c(28.67, 26.89, 27.82, 29.38, 27.53, 25.65, 26.73, 27.34, 25.99, 26.24) Based on this sample, find the 95% confidence interval for the true concentration u of the bacteria in millions per ml of fluid (i.e. in lOsfmI}. Do not round your answer. Create R vector CI of length two, so that the first and the second entries are the lower and the upper end points of your confidence interval interval. The values should be in millions per mi (L0%/mi). Hint: Make sure you have everything in millions per ml. Do not round your answer! # your code here fail{) # No Answer - remove if you provide an answer ## check whether CI is a numeric vector of length 2 if (test_that(desc="", code={ expect_equal( is.numeric(CI), TRUE) }) 1= TRUE) stop("Sorry, CI is not a numeric vector.") if (test_that(desc="", code={ expect_equal( length(CI)==2, TRUE) }) != TRUE) stop("Sorry, CI is not of length 2.") ## check whether whether 7th and 8th dec. digits of CI[1] and CI[2] are 45 and 54, respectively if (test_that(desc="", code={ expect_equal( (floor(CI[1]*10"8) %% 1@@ == 45), TRUE) }) != TRUE) stop("Sorry, your CI is incorrect.") if (test_that(desc="", code={ expect_equal( (floor(CI[2]%108) %% 100 == 54), TRUE) }) !'= TRUE) stop("Sorry, your CI is incorrect.") ## check whether the answer is correct (hidden test) Part (b) Find z > 0 such that P(| Z| Do not round your answer! In [ ]: # your code here fail() # No Answer - remove if you provide an answer In [ ]: ## check whether the 6th and 7th decimal digits of z are 36 if (test_that(desc="", code={ expect_equal( (floor(z#107) %% 100 == 36), TRUE) }) !'= TRUE) stop("Sorry, z is incorrect.") In [ ]: ## check whether the answer is correct (hidden tests) In [ 1: In [ ]: In [ ]: In [ 1: Part (c) How many measurements should you make to claim with at least 90% confidence that the obtained sample mean X, when used as an estimate of u, falls within margin of error less than 0.5 - 10 bacteria/ml from the true value M7 In other words, what should be the smallest sample size # for which P(|X pl