Question

1. dogs the number of dogs per household in a small town directions: use the frequency distribution to construct a probability distribution. then, find the mean and standard deviation of the probability distribution.

dogs\thousehold
0\t1491
1\t425
2\t168
3\t48
4\t29
5\t14

Explanation:

Step 1: Calculate Total Number of Households

First, we sum up the number of households for each number of dogs. Let \( N \) be the total number of households.
\( N = 1491 + 425 + 168 + 48 + 29 + 14 \)
\( N = 1491+425 = 1916 \); \( 1916 + 168 = 2084 \); \( 2084+48 = 2132 \); \( 2132 + 29 = 2161 \); \( 2161+14 = 2175 \)

Step 2: Calculate Probabilities (\( P(x) \))

For each \( x \) (number of dogs), \( P(x)=\frac{\text{Households with } x \text{ dogs}}{N} \)

• \( P(0)=\frac{1491}{2175} \approx 0.6855 \)
• \( P(1)=\frac{425}{2175} \approx 0.1954 \)
• \( P(2)=\frac{168}{2175} \approx 0.0772 \)
• \( P(3)=\frac{48}{2175} \approx 0.0221 \)
• \( P(4)=\frac{29}{2175} \approx 0.0133 \)
• \( P(5)=\frac{14}{2175} \approx 0.0064 \)

Step 3: Calculate Expected Value (\( \mu \))

The formula for the expected value of a discrete random variable is \( \mu=\sum_{i} x_i P(x_i) \)
\( \mu = 0\times0.6855 + 1\times0.1954 + 2\times0.0772 + 3\times0.0221 + 4\times0.0133 + 5\times0.0064 \)
\( \mu = 0 + 0.1954 + 0.1544 + 0.0663 + 0.0532 + 0.032 \)
\( \mu = 0.1954+0.1544 = 0.3498 \); \( 0.3498+0.0663 = 0.4161 \); \( 0.4161+0.0532 = 0.4693 \); \( 0.4693+0.032 = 0.5013 \)

Step 4: Calculate Variance (\( \sigma^2 \))

The formula for variance is \( \sigma^2=\sum_{i} (x_i - \mu)^2 P(x_i) \)

• For \( x = 0 \): \( (0 - 0.5013)^2\times0.6855 = (-0.5013)^2\times0.6855 \approx 0.2513\times0.6855 \approx 0.1723 \)
• For \( x = 1 \): \( (1 - 0.5013)^2\times0.1954 = (0.4987)^2\times0.1954 \approx 0.2487\times0.1954 \approx 0.0486 \)
• For \( x = 2 \): \( (2 - 0.5013)^2\times0.0772 = (1.4987)^2\times0.0772 \approx 2.2461\times0.0772 \approx 0.1734 \)
• For \( x = 3 \): \( (3 - 0.5013)^2\times0.0221 = (2.4987)^2\times0.0221 \approx 6.2435\times0.0221 \approx 0.1379 \)
• For \( x = 4 \): \( (4 - 0.5013)^2\times0.0133 = (3.4987)^2\times0.0133 \approx 12.2410\times0.0133 \approx 0.1628 \)
• For \( x = 5 \): \( (5 - 0.5013)^2\times0.0064 = (4.4987)^2\times0.0064 \approx 20.2383\times0.0064 \approx 0.1295 \)

Now sum these up: \( 0.1723 + 0.0486 + 0.1734 + 0.1379 + 0.1628 + 0.1295 \)
\( 0.1723+0.0486 = 0.2209 \); \( 0.2209+0.1734 = 0.3943 \); \( 0.3943+0.1379 = 0.5322 \); \( 0.5322+0.1628 = 0.695 \); \( 0.695+0.1295 = 0.8245 \)

Step 5: Calculate Standard Deviation (\( \sigma \))

Standard deviation is the square root of variance: \( \sigma=\sqrt{\sigma^2}=\sqrt{0.8245} \approx 0.908 \)

Answer:

The mean (expected value) is approximately \( 0.50 \) and the standard deviation is approximately \( 0.91 \) (values may vary slightly due to rounding during calculations). If we use more precise calculations:

• Mean: \( \mu \approx 0.5013 \approx 0.50 \)
• Standard Deviation: \( \sigma \approx \sqrt{0.8245} \approx 0.908 \approx 0.91 \)