SOLUTION: The U.S. Bureau of Labor Statistics reports that the average annual expenditure on food and drink for all families is $5700 (Money, December 2003). Assume that annual expenditure

Question 1180571: The U.S. Bureau of Labor Statistics reports that the average annual expenditure on food
and drink for all families is $5700 (Money, December 2003). Assume that annual expenditure
on food and drink is normally distributed and that the standard deviation is $1500.
a. What is the range of expenditures of the 10% of families with the lowest annual spending
on food and drink?
b. What percentage of families spend more than $7000 annually on food and drink?
c. What is the range of expenditures for the 5% of families with the highest annual spending
on food and drink?
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to solve this problem:
**a. Range of expenditures for the lowest 10%:**
1. **Find the z-score:** We're looking for the 10th percentile of the distribution. Use a z-table or calculator to find the z-score that corresponds to 0.10 (or 10%) of the area to the *left* of the mean. This z-score is approximately -1.28.
2. **Convert to expenditure:** Use the z-score formula to convert this to a dollar amount:
x = μ + zσ
x = $5700 + (-1.28)($1500)
x = $5700 - $1920
x = $3780
So, the range of expenditures for the lowest 10% of families is from $0 to $3780.
**b. Percentage of families spending more than $7000:**
1. **Calculate the z-score:**
z = (x - μ) / σ
z = ($7000 - $5700) / $1500
z = $1300 / $1500
z ≈ 0.87
2. **Find the probability:** Use a z-table or calculator to find the area to the *right* of z = 0.87. This represents the percentage of families spending more than $7000. P(z > 0.87) is approximately 0.1922 or 19.22%.
**c. Range of expenditures for the highest 5%:**
1. **Find the z-score:** We're looking for the 95th percentile (since we want the top 5%). Use a z-table or calculator to find the z-score that corresponds to 0.95 (or 95%) of the area to the *left* of the mean. This z-score is approximately 1.645.
2. **Convert to expenditure:**
x = μ + zσ
x = $5700 + (1.645)($1500)
x = $5700 + $2467.50
x = $8167.50
So, the range of expenditures for the highest 5% of families is from $8167.50 and above.
**Answers:**
* a. The range of expenditures for the lowest 10% of families is from $0 to $3780.
* b. Approximately 19.22% of families spend more than $7000 annually on food and drink.
* c. The range of expenditures for the highest 5% of families is from $8167.50 and above.

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