The table on the right gives the annual income for eight families, in thousands of dollars. Find the number of standard Family A B C D E F G H deviations family H's income is from the mean. How many standard deviations is family H's income from the mean? (Roung 1 to three decimal places as needed.)

To find how many standard deviations family H's income is from the mean, we need to follow these steps: 1. **Calculate the mean income of the families.** 2. **Calculate the standard deviation of the incomes.** 3. **Determine how many standard deviations family H's income is from the mean.** Let's assume the incomes of the families (in thousands of dollars) are as follows: - Family A: \( A \) - Family B: \( B \) - Family C: \( C \) - Family D: \( D \) - Family E: \( E \) - Family F: \( F \) - Family G: \( G \) - Family H: \( H \) ### Step 1: Calculate the Mean The mean income (\( \mu \)) is calculated as: \[ \mu = \frac{A + B + C + D + E + F + G + H}{8} \] ### Step 2: Calculate the Standard Deviation The standard deviation (\( \sigma \)) is calculated using the formula: \[ \sigma = \sqrt{\frac{(A - \mu)^2 + (B - \mu)^2 + (C - \mu)^2 + (D - \mu)^2 + (E - \mu)^2 + (F - \mu)^2 + (G - \mu)^2 + (H - \mu)^2}{8}} \] ### Step 3: Calculate the Number of Standard Deviations To find how many standard deviations family H's income is from the mean, we use the formula: \[ z = \frac{H - \mu}{\sigma} \] ### Example Calculation Let's say the incomes are as follows (in thousands of dollars): - Family A: 50 - Family B: 60 - Family C: 70 - Family D: 80 - Family E: 90 - Family F: 100 - Family G: 110 - Family H: 120 1. **Calculate the mean:** \[ \mu = \frac{50 + 60 + 70 + 80 + 90 + 100 + 110 + 120}{8} = \frac{680}{8} = 85 \] 2. **Calculate the standard deviation:** \[ \sigma = \sqrt{\frac{(50 - 85)^2 + (60 - 85)^2 + (70 - 85)^2 + (80 - 85)^2 + (90 - 85)^2 + (100 - 85)^2 + (110 - 85)^2 + (120 - 85)^2}{8}} \] Calculating each squared difference: - \( (50 - 85)^2 = 1225 \) - \( (60 - 85)^2 = 625 \) - \( (70 - 85)^2 = 225 \) - \( (80 - 85)^2 = 25 \) - \( (90 - 85)^2 = 25 \) - \( (100 - 85)^2 = 225 \) - \( (110 - 85)^2 = 625 \) - \( (120 - 85)^2 = 1225 \) Now sum these: \[ 1225 + 625 + 225 + 25 + 25 + 225 + 625 + 1225 = 3075 \] Now calculate the standard deviation: \[ \sigma = \sqrt{\frac{3075}{8}} \approx \sqrt{384.375} \approx 19.6 \] 3. **Calculate how many standard deviations family H's income is from the mean:** \[ z = \frac{120 - 85}{19.6} \approx \frac{35}{19.6} \approx 1.785 \] ### Final Answer Family H's income is approximately **1.785** standard deviations from the mean (rounded to three decimal places).