7454 B O1xythos.content.blackboardcdn.com C), less than 7,500 use the app for less than 18 minutes? Hint: You'll want to approximate a binomial distribution with LL normal distribution here. 20f2 MATH 177, 2021 Assignment 6 4 Iron ore is dug up to be crushed and ltered through a sieve prior to further processing. The sieve lters out any iron ore larger than 1cm in diameter. Let X be a random variable describing the diameter (in cm) of a piece that has made it through the sieve. Suppose that X has the beta distribution with parmncters a = 4 and {i = 3. (a) Determine the cdf of X. (b) Calculate the mean and standard deviation of the diameter of iron ore pieces. (c) The dollar value V of an iron ore piece is proportional to its volume, specically V = $ 5X3. Find E[V], the average dollar value of a piece that makes it through the sieve. 5 Let X be a continuous random variable with cumulative distribution function Consider a second random variable described by Y = 6/}. (a) Determine the mean and median of X. (b) What interval do samples of Y lie in? (c) Determine P(Y S y) for y lying in this interval. (:1) Thus, specify Fy(y) in full and conclude what family of continuous distributions Y belongs to. 6 A team of conservationists capture and measure the weight of 100 adult New Zealand kaka. The results are set out in the following table. Weight Numb er of kakas (a) (frequency) 390-410 7 410-430 16 430-450 23 450-470 17 470-490 11 490-510 9 510-530 6 530-550 6 550-570 5 (a) Find the sample mean weight a": and standard deviation 8. (b) Assuming that i: and s are good approximations to H and a, fit a normal distribution to this data by obtaining the expected frequencies, to M decimal places. (c) Perform a goodnessoft test for the normal model and make a conclusion regarding the suitability of the model to describe this data. MATH 177, 2021 Assignment 6 7454 B I) xythos.content.blackboardcdn.com C), 1of2 SCHOOL or MATHEMATICS AND STATISTICS Te Kara Matai Tataamnga MATH 177 Assigrunent 6 Due: 11:59pm, 7 October 2021 Note: Make sure you transform to the standard normal random variable when calculating prob- abilities associated with any normal random variablesY there will be marks associated with this! 1 A pathologist uses a microuidic device to separate circulating tumor cells (CTCs) from a blood sample and then measures their diameter. Let X be the diameter in microns (pm) of a CTC produced by a certain type of cancer. Suppose X has the cdf 0 r: 16 . (a) What is the probability that such a CTC has diameter greater than 12 microns? (b) Determine a suitable probability density function for X. (c) What is the mean and standard deviation of CTC diameters? 2 An arborist collects data on the height of mature rimu trerS in the local area. Letting X be the height in metres of a randomly selected rhnu tree, then the data suggests that P (X > 35) = 0.08 and P(X