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, kWh
- Standard deviation, kWh
- Lower bound, kWh
- Upper bound, 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: For kWh: For kWh:
Step 2: Find the probabilities
Using the standard normal distribution table or a calculator:
- The probability corresponding to is approximately 0.5910.
- The probability corresponding to 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:
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:
- How do you interpret the result of a Z-score?
- What is the probability that energy consumption exceeds 1225 kWh?
- How do you find the probability that energy consumption is less than 1100 kWh?
- Can you calculate the probability for consumption between 1000 kWh and 1300 kWh?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Statistics
Normal Distribution
Z-scores
Formulas
Z = (X - μ) / σ
Theorems
Standard Normal Distribution Theorem
Suitable Grade Level
Grades 10-12
Related Recommendation
Find the 45th Percentile of a Normal Distribution with Mean 1050 kWh and Standard Deviation 218 kWh
Normal Distribution: Calculating Percentage of Homes with Utility Bills between $91 and $143
Calculate Percentage of Homes with Utility Bills in a Normal Distribution
Normal Distribution Problem: Percentage of Homes with Utility Bill over $104
Calculating Probabilities Using Normal Distribution with Mean $100 and Standard Deviation $13