Question

In 1940 the average size of a U.S. farm was 174 acres. Let's say that the standard deviation was 59 acres. Suppose we randomly survey 39 farmers from 1940.

I.   In words, define the random variable X.

II. In words, define the random variable X-bar?

III. Give the distribution of X-bar. (Round your standard deviation to 2 decimal places)

X ~ ____ ( ____, ____ ).

IV. The middle 50% of the distribution for X-bar, the bounds of which form the distance represented by the IQR, lies between what two values? (Round your answers to two decimal places.)

_____ acres (smaller value)

_____ acres (larger value)

List your process according to the central limit theorem.

Answer 1

i)

here X is size of farm for a randomely selected farmer.

ii)Xbar is average farm size for 39 randomely selected farmers

iii)from central limit theorum as sample size is greater than 30; tehrefore sample mean will follow normal distribution with mean =174

and std error of mean =59/sqrt(39)=9.45

Xbar ~ (174 ,9.45)

iv)for middle 50% values crtiical z =-/+0.67

hence smaller value =mean+z*Std deviaiton ==174-0.67*9.45=167.67

larger value =mean+z*Std deviaiton ==174+0.67*9.45=180.33