Question

66. In 1940 the average size of a U.S. farm was 174 acres. Let's say that the standard deviation was 55 acres. Suppose we randomly survey 38 farmers from 1940.

a. In words, X=

b. In words, $\stackrel{-}{X}$=

c. $\stackrel{-}{X~}$

d. The IQR for $\stackrel{-}{X}$is from acres to acres.

Short answer

a. X is the size of the US farm

b. $\stackrel{-}{X}$ is the average size

c.$\stackrel{-}{X}~N(174,8.922)$

d.$168-180$acres

Step-by-step solution

Given Information

A random variable is a variable whose potential qualities are mathematical results of an irregular peculiarity. There are two kinds of irregular factors, discrete and nonstop.

Explanation Part (1)

a. The size of the US farms is defined as X

b. The average size of the US farm after surveying $38$farms

c. Given,

The standard deviation $\sigma$=$55$

Sample size n = $38$

and Mean $\mu =174$

From the formula,

$$\begin{gathered} \overline{=X}~N\left({\mu }_x,\frac{{\sigma }_x}{\sqrt{n}}\right) \\ =\overline{X}~N\left(174,\frac{55}{\sqrt{38}}\right) \\ =\overline{X}~N(174,8.922) \end{gathered}$$

Explanation Part (2)

d. Given,

The standard deviation $\sigma =55$

Sample size n =$38$

and Mean = $\mu =174$

$\overline{X}~N(174,8.922)$

First quartile is$25th$percentile

Hence, $P(\overline{x}<k)=0.25$

The value of k using a calculator is,$168$acres

The third quartile is $75th$percentile

Hence, $P(\overline{x}<k)=0.75$

The value of k using a calculator is, $\text{invnorm}(0.75,174,8.922)=180$acres

Thus the IQR is from$168-180$acres