Select the correct answer.
The monthly electric bills in a certain community are normally distributed, with a mean of $245 and a standard distribution of $55. If there are 461 households in the community, about how many of them have electric bills less than $200?
A. 81
B. 45
C. 21
D. 95
Answer (1)
Calculate the z-score: z = 55 200 − 245 = − 0.82
Find the area to the left of z = − 0.82 in the z-table: 0.20611.
Multiply the area by the total number of households: 0.20611 × 461 = 94.99671 .
Round to the nearest whole number: 95 .
Explanation
Understand the problem and provided data We are given that the monthly electric bills in a community are normally distributed with a mean of $245 and a standard deviation of $55. We want to find the number of households out of 461 that have electric bills less than $200.
Calculate the z-score First, we need to calculate the z-score for a bill of $200. The z-score formula is: z = σ x − μ where x is the value, μ is the mean, and σ is the standard deviation. In this case, x = 200 , μ = 245 , and σ = 55 .
Compute the z-score Plugging in the values, we get: z = 55 200 − 245 = 55 − 45 = − 0.818181... Rounding to two decimal places, we have z = − 0.82 .
Find the area to the left of z in the table Now, we look up the area to the left of this z-score in the z-table. For z = − 0.82 , the area to the left is approximately 0.20611. This represents the proportion of households with bills less than $200.
Estimate the number of households To find the number of households with bills less than $200, we multiply the area (probability) by the total number of households (461): 0.20611 × 461 = 94.99671 Rounding to the nearest whole number, we get approximately 95 households.
State the final answer Therefore, about 95 households have electric bills less than $200.
Examples
Understanding normal distributions and z-scores can help in various real-life scenarios. For example, a teacher can analyze student test scores to see how many students performed above or below average. Similarly, a business owner can analyze sales data to understand how many products fall within a certain price range. In this case, we determined that approximately 95 households have electric bills less than $200, which can help the electric company understand the distribution of bills in the community. The z-score calculation is: z = σ x − μ , and the estimated number of households is calculated by multiplying the probability (area to the left of z) by the total number of households.
