Math Problem Statement
We are given a normally distributed random variable X with µ = 75 and σ = 5.
Find P (X > 85).
Find k such as P(X<k) = 0.997. Interpret your result.
Find the two values where the middle 68.27% of the distribution of X lie.
Solution
We are given that the random variable is normally distributed with and , meaning:
1. Find
We first standardize the value of 85 by converting it to the standard normal form , where:
So, for :
Now, we need to find .
Using standard normal tables or a calculator:
Thus:
2. Find such that
We need to find the value of such that . This means finding the corresponding -value for which the cumulative probability is 0.997.
Using standard normal tables, we find that:
Now, convert the -value back to using:
So:
Thus, the value of is approximately 88.74. This means that 99.7% of the distribution of lies below 88.74.
3. Find the two values where the middle 68.27% of the distribution lies
The middle 68.27% of the distribution corresponds to one standard deviation above and below the mean in a normal distribution.
For the standard normal distribution, this means:
Thus:
So, the middle 68.27% of the distribution lies between 70 and 80.
Summary of Results:
- such that
- The middle 68.27% of the distribution lies between 70 and 80.
Would you like more details or clarification on any part of this?
Here are some related questions to expand on this topic:
- How do you compute probabilities for other values of ?
- What is the 95th percentile for this distribution?
- How do changes in and affect the distribution?
- How can you interpret the result for in practical terms?
- What is the probability that is between 80 and 85?
Tip: When dealing with normal distributions, always standardize using the formula to simplify the calculations!
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Math Problem Analysis
Mathematical Concepts
Statistics
Probability
Normal Distribution
Standard Normal Distribution
Formulas
Z = (X - µ) / σ
P(X > 85) = P(Z > 2)
k = Z * σ + µ
Theorems
68-95-99.7 Rule (Empirical Rule)
Suitable Grade Level
College-level Statistics