Suppose X is a normally distributed random variable with the mean u(Mu) and variance sigma^2. Suppose P(X </= 10) - .6915 and P (X </= 20) -.8413. Find Mu and Sigma^2.
PLEASE HELP! THIS IS AN EXAM REVIEW QUESTION!
for z-scores,
z = (x - m)/s
from my charts, a prob of .6915 corresponds to a z-score of .5
so .5 = (10 - m)/s
.5s = 10-m
m = 10 - .5s
and a prob of .8413 corresponds to a z-score of .9998 or 1
1 = (20-m)/s
s= 20-m
m = 20 - s
then 20 - s = 10 - .5s
s = -10/-.5 = 20
m = 0
If I recall variance is the square of the standard deviation.
I checked my answers with
http://davidmlane.com/hyperstat/z_table.html
To find the values of the mean (µ) and variance (σ^2), we need to use the standard normal distribution table and the properties of the normal distribution.
First, let's understand what the given probabilities mean. P(X ≤ 10) = 0.6915 and P(X ≤ 20) = 0.8413.
To find the mean (µ), we need to determine the corresponding z-score for each probability. The z-score represents the number of standard deviations away from the mean a particular value is.
From the standard normal distribution table, you can find the z-score that corresponds to a given probability.
For P(X ≤ 10) = 0.6915, the corresponding z-score is approximately 0.511.
For P(X ≤ 20) = 0.8413, the corresponding z-score is approximately 0.994.
Now, we can use the z-score formula to find the values of the mean (µ) and variance (σ^2):
z = (X - µ) / σ
Since X is normally distributed with mean µ and variance σ^2, we can rewrite the probabilities as:
P(Z ≤ 0.511) = 0.6915
P(Z ≤ 0.994) = 0.8413
By substituting the z-scores into the equation, we can solve for µ and σ.
For the first equation:
0.511 = (10 - µ) / σ
For the second equation:
0.994 = (20 - µ) / σ
Solving these two equations simultaneously will give us the values of µ and σ.
Let's rearrange the first equation to solve for (10 - µ):
(10 - µ) = 0.511σ
Now substitute (10 - µ) into the second equation:
0.994 = (20 - (10 - µ)) / σ
Simplify:
0.994 = (20 - 10 + µ) / σ
0.994 = (10 + µ) / σ
Cross-multiply:
0.994σ = 10 + µ
Now we have two equations:
(10 - µ) = 0.511σ ---(equation 1)
0.994σ = 10 + µ ---(equation 2)
Next, we can solve these two equations simultaneously. You can solve them algebraically or using a numerical method.
By solving these equations, you will find the values of the mean (µ) and variance (σ^2) that make the given probabilities true.
Note: The exact values of µ and σ^2 will depend on the specific calculations.