Answer to Question #350756 in Statistics and Probability for Muhammad hassan

Question

A market researcher at a major company classified households by car ownership. The relative frequencies of households for each category of ownership are shown in Table 1.

Table 1. The relative frequencies of households

Number of cars House hold 0 1 2 3 4 5

Relative Frequency 0.1 0.3 0.4 0.12 0.06 0.02

Calculate the mean value and standard deviation of the random variable and interpret the result.

Answer 1

$$mean=\mu=E(X)=0.1(0)+ 0.3(1)+ 0.4(2)$$

$$+ 0.12(3)+ 0.06(4)+ 0.02(5)=1.8$$

$$E(X^2)=0.1(0)^2+ 0.3(1)^2+ 0.4(2)^2$$

$$+ 0.12(3)^2+ 0.06(4)^2+ 0.02(5)^2=4.44$$

$$Var(X)=\sigma^2=E(X^2)-(E(X))^2$$

$$=4.44-(1.8)^2=1.2$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{1.2}\approx1.0954$$

A large standard deviation (or variance) means that the distribution is spread out, with some chance of observing values at some distance from the mean.