Question

Complete the following problems using R.  Name the file with your last name, e.g., “Prob set 6-Smith.eoc”Clearly label problems, and be sure to turn in explanations and interpretations where appropriate.

1.Lymphocytes are a type of white blood cell found in bone marrow and lymph tissue, and a main type of immune cells. Lymphocyte levels in 1 microliter (µL) of blood are normally distributed with a mean of 2300 and standard deviation 250 mm.

a.     What is the probability that a randomly selected patient will have a lymphocyte level between 2110 and 2300 (per microliter (µL) of blood)?

b.     What is the probability that the average lymphocyte level from 10 patients will be between 2110 and 2300?

c.     What is the probability that the average lymphocyte level from 20 patients will be between 2110 and 2300?

How are your answers from (a), (b) and (c) different, and why?

Answer 1

from R:

pnorm command will give above probability

here for pnorm(x,mean,sd) paramter mean=2300 and sd =250

a)

> pnorm(2300,2300,250)-pnorm(2110,2300,250)
[1] 0.2763727

b)

here std error of mean =std deviaiton/sqrt(n)=250/sqrT(10)

> pnorm(2300,2300,250/sqrt(10))-pnorm(2110,2300,250/sqrt(10))
[1] 0.4918768

c)

> pnorm(2300,2300,250/sqrt(20))-pnorm(2110,2300,250/sqrt(20))
[1] 0.4996616

here as we increase sample size frm a-c ; the probability increases as with increae in sample size ; standard error of mean reduces and sample mean is more inclined to be closer to population mean