Suppose a simple random sample of size nequals36 is obtained from a population with mu equals 74 and sigma equals 6. (a) Describe the sampling distribution of x overbar. (b) What is Upper P (x overbar greater than 75.9 )? (c) What is Upper P (x overbar less than or equals 71.95 )? (d) What is Upper P (73 less than x overbar less than 75.75 )?

Part a)

The simple random sample of size n=36 is obtained from a population with

and

The sampling distribution of the sample means has a mean that is equal to mean of the population the sample has been drawn from.

Therefore the sampling distribution has a mean of

The standard error of the means becomes the standard deviation of the sampling distribution.

Part b) We want to find

We need to convert to z-score.

Part c)

We want to find

We convert to z-score and use the normal distribution table to find the corresponding area.

Part d)

We want to find :

We convert to z-scores and again use the standard normal distribution table.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!