In a population of exam scores,a score of X= 48 corresponds to z= 1 and a score of X= 36 corresponds to z= -.50. Find the mean and standard deviation for the population?
z-score = (x-mean)/sd
1 = (48-m)/s
s = 48-m
s+m = 48 , #1
-1/2 = (36-m)/s
(-1/2)s = 36-m
-s = 72 - 2m
2m - s = 72 , #2
add #1 and #2
3m = 120
m = 40
then s+40 = 48
s = 8
mean = 40 , SD = 8
To find the mean and standard deviation for the population, we need to use the standard score equation.
The standard score, also known as the z-score, represents how many standard deviations a particular data point is from the mean. It is calculated using the formula:
z = (X - μ) / σ
Where:
- z is the standard score
- X is the data point
- μ is the population mean
- σ is the population standard deviation
Given that a score of X = 48 corresponds to z = 1, we can substitute these values into the standard score equation:
1 = (48 - μ) / σ
Similarly, for X = 36 and z = -0.50:
-0.50 = (36 - μ) / σ
Now we have a system of two equations with two unknowns (μ and σ). We can solve these equations simultaneously to find the values of μ and σ.
Let's solve the first equation for μ:
1 = (48 - μ) / σ
σ = 48 - μ (multiply both sides by σ)
Substitute this value of σ into the second equation:
-0.50 = (36 - μ) / (48 - μ)
Now solve for μ:
-0.50(48 - μ) = 36 - μ
-24 + 0.50μ = 36 - μ
1.50μ = 60
μ = 60 / 1.50
μ = 40
Now substitute this value of μ back into the equation for σ:
σ = 48 - μ
σ = 48 - 40
σ = 8
Therefore, the mean (μ) for the population is 40, and the standard deviation (σ) is 8.