In a distribution of scores, X=62 corresponds to z=+0.50, and X=52 corrsponds to z=-2.00. Find the mean and standard deviation for the distribution.
To find the mean and standard deviation for the distribution, we can use the formulas that relate the z-score to the corresponding value in the distribution.
The formula to convert a value of X in the distribution to its corresponding z-score is:
z = (X - μ) / σ
Where:
- z is the z-score,
- X is a value in the distribution,
- μ is the mean of the distribution, and
- σ is the standard deviation of the distribution.
We are given two points of the distribution:
1. X = 62, z = +0.50
2. X = 52, z = -2.00
Step 1: Finding the mean (μ)
Using the formula for z-score:
0.50 = (62 - μ) / σ
Rearranging the equation:
62 - μ = 0.50σ ----(1)
Step 2: Finding the standard deviation (σ)
Using the formula for z-score:
-2.00 = (52 - μ) / σ
Rearranging the equation:
52 - μ = -2.00σ ----(2)
Step 3: Solve the equations
To find the values of μ and σ, we need to solve the system of equations (1) and (2).
From equation (1):
62 - μ = 0.50σ
From equation (2):
52 - μ = -2.00σ
We can solve this system of equations using substitution or elimination.
Substituting the value of σ from equation (1) into equation (2):
52 - μ = -2.00(62 - μ) / 0.50
Simplifying:
52 - μ = -4(62 - μ)
Distributing -4:
52 - μ = -248 + 4μ
Combining like terms:
5μ = 300
Dividing both sides by 5:
μ = 60
Now we have found the mean of the distribution, which is μ = 60.
Step 4: Finding the standard deviation
Using equation (1):
62 - μ = 0.50σ
Replacing the value of μ:
62 - 60 = 0.50σ
2 = 0.50σ
Dividing both sides by 0.50:
4 = σ
Now we have found the standard deviation of the distribution, which is σ = 4.
Therefore, the mean of the distribution is 60 and the standard deviation is 4.
62 - 52 = 10
10 = 2.5 SD (from -2 to +.5)
SD = 4
Z = (X - μ)/SD
Insert one value to find μ, and use other other value to check.