To construct a probability distribution for the number of televisions per household, follow these detailed steps:
### Step 1: Identify the given data
The given data consists of the number of televisions per household and the corresponding number of households:
- Number of televisions (X): 0, 1, 2, 3
- Number of households (N): 28, 446, 726, 1408
### Step 2: Compute the total number of households
First, we need to find the total number of households by summing up the given households:
[tex]\[
\text{Total households} = 28 + 446 + 726 + 1408
\][/tex]
Summing these values gives us:
[tex]\[
\text{Total households} = 2608
\][/tex]
### Step 3: Calculate the probability for each number of televisions
To find the probability of each event (number of televisions), divide the number of households with that many televisions by the total number of households. The probability [tex]\( P(X = x) \)[/tex] for each [tex]\( x \)[/tex] can be calculated as follows:
[tex]\[
P(X = 0) = \frac{28}{2608}
\][/tex]
[tex]\[
P(X = 1) = \frac{446}{2608}
\][/tex]
[tex]\[
P(X = 2) = \frac{726}{2608}
\][/tex]
[tex]\[
P(X = 3) = \frac{1408}{2608}
\][/tex]
### Step 4: Round the probabilities to 3 decimal places
We need to round the calculated probabilities to three decimal places for each number of televisions. Performing the division and rounding, we get:
[tex]\[
P(X = 0) \approx 0.011
\][/tex]
[tex]\[
P(X = 1) \approx 0.171
\][/tex]
[tex]\[
P(X = 2) \approx 0.278
\][/tex]
[tex]\[
P(X = 3) \approx 0.540
\][/tex]
### Step 5: Present the probability distribution
The final probability distribution for the number of televisions per household is:
[tex]\[
\begin{array}{c|c}
\text{Number of Televisions (X)} & \text{Probability} \, P(X = x) \\
\hline
0 & 0.011 \\
1 & 0.171 \\
2 & 0.278 \\
3 & 0.540 \\
\end{array}
\][/tex]
Each probability represents the fraction of the total 2608 households that have 0, 1, 2, and 3 televisions, respectively. Each of these probabilities has been rounded to three decimal places.
