Question

complete parts (a) and (b) below
the number of dogs per household in a small town
dogs 0 1 2 3 4 5
probability 0.655 0.214 0.084 0.023 0.015 0.009
find the variance of the probability distribution
σ² = 0.9 (round to one decimal place as needed.)
find the standard deviation of the probability distribution
σ = 0.9 (round to one decimal place as needed.)
(b) interpret the results in the context of the real - life situation.
○ a. a household on average has 0.6 dog with a standard deviation of 1.0 dog
○ b. a household on average has 1.0 dog with a standard deviation of 0.6 dog
○ c. a household on average has 0.6 dog with a standard deviation of 17 dogs
○ d. a household on average has 0.9 dog with a standard deviation of 1.0 dog.

Explanation:

Step1: Calculate the mean ($\mu$)

$$\mu=\sum_{x}x\cdot P(x)$$
$$\mu = 0\times0.655+1\times0.214 + 2\times0.084+3\times0.023+4\times0.015+5\times0.009$$
$$\mu=0 + 0.214+0.168 + 0.069+0.06+0.045$$
$$\mu = 0.556\approx0.6$$

Step2: Recall the relationship between variance ($\sigma^{2}$) and standard deviation ($\sigma$)

We know that $\sigma=\sqrt{\sigma^{2}}$. Given $\sigma^{2} = 0.9$ (rounded), then $\sigma=\sqrt{0.9}\approx0.9487\approx0.9\approx1.0$ (when considering the options, likely a rounding - related context in the problem's setup for the options).

Answer:

A. A household on average has 0.6 dog with a standard deviation of 1.0 dog