Respuesta :
Answer:
a. What value will be exceeded 75% of the time?
= 21.63
b. What value will be exceeded 10% of the time?
= 31.41
c. What value will be exceeded 85% of the time?
= 19.82
d. Determine two values of which the smallest has 25% of the values below it and the largest has 25% of the values above it.
The two values are: 21.63 and 28.37
Step-by-step explanation:
We solve using the z score formula:
z = (x-μ)/σ, where
x is the raw score
μ is the population mean
σ is the population standard deviation.
A random variable is normally distributed with a mean of 25 and a standard deviation of 5. If an observation is randomly selected from the distribution.
Required:
a. What value will be exceeded 75% of the time?
We are to find the value x
Exceeded by 75% = 100 - 75 = 25
We find the z score of 25% percentile
= -0.674
Hence:
-0.674 = x - 25/5
Cross Multiply
-0.674 × 5 = x - 25
-3.37 = x - 25
x = 25 - 3.37
x = 21.63
b. What value will be exceeded 10% of the time?
We are to find the value x
Exceeded by 90% = 100 - 90 = 10
We find the z score of 90% percentile
= 1.282
Hence:
1.282 = x - 25/5
Cross Multiply
1.282 × 5 = x - 25
6.41 = x - 25
x = 6.41 + 25
x = 31.41
c. What value will be exceeded 85% of the time?
We are to find the value x
Exceeded by 85% = 100 - 85 = 15
We find the z score of 15% percentile
= -1.036
Hence:
-1.036 = x - 25/5
Cross Multiply
-1.036 × 5 = x - 25
-5.18 = x - 25
x = 25 - 5.18
x = 19.82
d. Determine two values of which the smallest has 25% of the values below it and the largest has 25% of the values above it.
This means their z score is 25% below the mean and 25% above the mean
We are to find the value x
We find the z score of 25 % and 75% percentile
= - 0.674 and 0.674
Hence:
For -0.674
-0.674 = x - 25/5
Cross Multiply
-0.674 × 5 = x - 25
-3.37 = x - 25
x = 25 - 3.37
x = 21.63
For 0.674
0.674 = x - 25/5
Cross Multiply
0.674 × 5 = x - 25
3.37 = x - 25
x = 25 + 3.37
x = 28.37
Therefore, the two values are: 21.63 and 28.37
