Questions: A frequency distribution is shown below. Complete parts (a) through (d). The number of televisions per household in a small town Televisions 0 1 2 3 Households 22 446 720 1400 (a) Use the frequency distribution to construct a probability distribution. x P(x) 0 square 1 square 2 square 3 square (Round to the nearest thousandth as needed.)

Solution Steps

The probability distribution for the number of televisions per household is calculated as follows:

\[ P(0) = \frac{22}{2200} = 0.009 \] \[ P(1) = \frac{446}{2200} = 0.172 \] \[ P(2) = \frac{720}{2200} = 0.278 \] \[ P(3) = \frac{1400}{2200} = 0.541 \]

Thus, the probability distribution is: \[ \begin{array}{ll} \mathbf{x} & \mathbf{P}(\mathbf{x}) \\ 0 & 0.009 \\ 1 & 0.172 \\ 2 & 0.278 \\ 3 & 0.541 \\ \end{array} \]

The mean \( \mu \) of the distribution is calculated using the formula:

\[ \mu = \sum (x \cdot P(x)) = 0 \times 0.009 + 1 \times 0.172 + 2 \times 0.278 + 3 \times 0.541 \]

Calculating this gives:

\[ \mu = 0 + 0.172 + 0.556 + 1.623 = 2.351 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \sum (P(x) \cdot (x - \mu)^2) \]

Calculating this gives:

\[ \sigma^2 = (0 - 2.351)^2 \times 0.009 + (1 - 2.351)^2 \times 0.172 + (2 - 2.351)^2 \times 0.278 + (3 - 2.351)^2 \times 0.541 \]

Calculating each term:

\[ = (5.528) \times 0.009 + (1.818) \times 0.172 + (0.123) \times 0.278 + (0.426) \times 0.541 \] \[ = 0.049752 + 0.312936 + 0.034194 + 0.227286 = 0.626 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{0.626} \approx 0.791 \]

Final Answer