Math Problem Statement

In one region, the September energy consumption levels for single -family homes are found to be normally distributed with a mean of 1050 kWh and a standard deviation of 218 kWh. For a randomly selected home, find the probability that the September energy consumption level is between 1100 kWh and 1225 kWh.

Solution

To find the probability that the September energy consumption level is between 1100 kWh and 1225 kWh, we can standardize the values and use the standard normal distribution (Z-distribution).

Given Data:

  • Mean, μ=1050\mu = 1050 kWh
  • Standard deviation, σ=218\sigma = 218 kWh
  • Lower bound, X1=1100X_1 = 1100 kWh
  • Upper bound, X2=1225X_2 = 1225 kWh

We will calculate the Z-scores for both values and then find the area between these Z-scores.

Step 1: Calculate the Z-scores

The Z-score formula is: Z=XμσZ = \frac{X - \mu}{\sigma} For X1=1100X_1 = 1100 kWh: Z1=11001050218=502180.2294Z_1 = \frac{1100 - 1050}{218} = \frac{50}{218} \approx 0.2294 For X2=1225X_2 = 1225 kWh: Z2=12251050218=1752180.8028Z_2 = \frac{1225 - 1050}{218} = \frac{175}{218} \approx 0.8028

Step 2: Find the probabilities

Using the standard normal distribution table or a calculator:

  • The probability corresponding to Z1=0.2294Z_1 = 0.2294 is approximately 0.5910.
  • The probability corresponding to Z2=0.8028Z_2 = 0.8028 is approximately 0.7888.

Step 3: Calculate the probability between the two Z-scores

The probability that the energy consumption is between 1100 kWh and 1225 kWh is the difference between the two probabilities: P(1100X1225)=P(Z2)P(Z1)=0.78880.5910=0.1978P(1100 \leq X \leq 1225) = P(Z_2) - P(Z_1) = 0.7888 - 0.5910 = 0.1978

Final Answer:

The probability that the energy consumption level is between 1100 kWh and 1225 kWh is approximately 0.1978, or 19.78%.

Would you like more details on any of these steps or further explanations?


Here are 5 related questions:

  1. How do you interpret the result of a Z-score?
  2. What is the probability that energy consumption exceeds 1225 kWh?
  3. How do you find the probability that energy consumption is less than 1100 kWh?
  4. Can you calculate the probability for consumption between 1000 kWh and 1300 kWh?
  5. How would the result change if the standard deviation were higher?

Tip: When using a Z-table, remember that it gives the cumulative probability up to a Z-score. Subtract to find probabilities between two Z-scores.