a) LetX ~ N(u = 2,0-2). Determine Paxsa). Answer b) i) Determine what percentage

Question

a) LetX ~ N(u = 2,0-2). Determine Paxsa). Answer b) i) Determine what percentage of observations would be expected to be outliers based on the 1.5 IQR rule (use qnorm and pnorm to perform the calculation). How many outliers would there expected to be in a sample of 100 observations from a normal distribution? Answer i Perform a simulation in R to substantiate the calculation in the previous part by sampling 100 observations at a time using the rnorm command. Perform 1,000 replicates and measure how many observations would be flagged based on the 1.5 IQR rule in each sample. Note: be sure to use the estimates of the quartiles and IQR from each sample rather than use the theoretical values from the previous part as this is what would be done in practice. Compare the average of these 1,000 replicates to what you would expect from the previous part, and explain in 3 or fewer sentences. Answer c) The town of Woburn, MA (a northern suburb of Boston) had an unusually high number of childhood leukemia cases from 1960 to 1986: there were in fact 28 such cases. The incidence rate of leukemia is about 1 in 5000 children per year. Assume the population of children in Woburn stayed constant at 2,400 children over this 27 year time period. Determine the probability of seeing 28 or more childhood leukemia cases in Woburn over this time period. Answer d) The genetics of Icelandic Sheep's wool color is complex, but a simplified model states that there are 3 main alleles for a single gene: black, white, and brown (sometimes called moorit) Black is dominant to brown and white and brown is dominant to white. The allele frequency in the general population is about 1/4 black, 1/4 brown, and 1/2 white. i) A ewe (female) named Bessie has a black coat. What is the probability that she is homozygous black? Answer: i) Bessie has been bred twice: once to a homozygous brown ram (male) and once to a homozygous white ram. Both times she gave birth to fraternal twin lambs that had black coats (a total of 4 black rams). Given this information, what is the probability that Bessie is homozygous black?

Explanation / Answer

(a) X ~ N(µ=2; = 2).

We need to find P(X2 > 4). This is equivalent to finding P(X>2) + P(X<-2)

P(X<-2) = 0.02275

P(X>2) = 1-P(X<2) = 0.5

So, P(X2 > 4) = P(X>2) + P(X<-2) = 0.02275 +0.5 = 0.52275

(c) The distribution follows a Poisson distribution, with as given below

Expected number of childhood leukemia cases in a period of 17 years with 2400 children and incidence rate of 1 case in 5000, = 12.96

We need to find the probability that X >=28 for this Poisson distribution with =12.96 can be calculated using Excel.

P(X<28) = POISSON.DIST(27,12.96,TRUE) = 0.999805

P(X>=28) = 1-0.999805 = 0.000195.

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