In a population of exam scores, a score of x=88 corresponds to z=+2.00 and a score of x=79 corresponds to z=-1.00. Find the mean and standard deviation for the population.
z/sigma = score - mean
2 sigma = 88 - mean
-sigma = 79 - mean
subtract second equation from first.
3 sigma = 9
sigma = 3 is the standard deviation
Use either equation to get the mean
To find the mean and standard deviation for the population, we can use the concept of z-scores.
The formula for finding the z-score is:
z = (x - μ) / σ,
where:
z is the z-score,
x is the individual score,
μ is the mean of the population, and
σ is the standard deviation of the population.
Given information:
x₁ = 88, z₁ = +2.00,
x₂ = 79, z₂ = -1.00.
Using the first set of information, we can write:
2.00 = (88 - μ) / σ,
and using the second set of information, we can write:
-1.00 = (79 - μ) / σ.
To solve these two equations, we need to eliminate μ so that we can solve for σ.
Rearranging the first equation, we get:
2.00σ = 88 - μ.
Now, substitute this expression into the second equation:
-1.00 = (79 - (2.00σ)) / σ.
Simplifying:
-1.00 = 79/σ - 2.00.
Now, cross-multiply:
-1.00σ = 79 - 2.00σ.
Collecting like terms:
1.00σ = 79.
Dividing both sides by 1.00:
σ = 79.
Substitute this value of σ into one of the previous equations to solve for μ. Let's use the first equation:
2.00 = (88 - μ) / 79.
Multiplying both sides by 79:
158 = 88 - μ.
Rearranging the equation:
μ = 88 - 158.
Simplifying:
μ = -70.
So, the mean of the population is μ = -70, and the standard deviation is σ = 79.
To find the mean and standard deviation for the population, we need to use the formula for converting a score to a z-score. The formula for converting a score to a z-score is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the score
- μ is the mean
- σ is the standard deviation
Using the given information, we can set up two equations:
For x = 88, z = +2.00
2 = (88 - μ) / σ
For x = 79, z = -1.00
-1 = (79 - μ) / σ
We now have a system of two equations with two variables (μ and σ). We can solve this system to find the mean (μ) and the standard deviation (σ).
Let's start by solving the first equation for μ in terms of σ:
2 = (88 - μ) / σ
Multiply both sides by σ:
2σ = 88 - μ
Subtract 88 from both sides:
2σ - 88 = -μ
Multiply both sides by -1 (to make μ positive):
μ = 88 - 2σ
Next, substitute this expression for μ into the second equation:
-1 = (79 - μ) / σ
-1 = (79 - (88 - 2σ)) / σ
-1 = (79 - 88 + 2σ) / σ
-1 = (-9 + 2σ) / σ
Multiply both sides by σ:
-σ = -9 + 2σ
Add σ to both sides:
0 = 9 + σ
Subtract 9 from both sides:
σ = -9
Oops! It seems like there might have been a mistake in the equation setup or solution process. Let's try again:
Let's solve the second equation for μ in terms of σ:
-1 = (79 - μ) / σ
Multiply both sides by σ:
-σ = 79 - μ
Add μ to both sides:
μ - σ = 79
Now substitute this expression for μ into the first equation:
2 = (88 - μ) / σ
2 = (88 - (μ - σ)) / σ
2 = (88 - μ + σ) / σ
Multiply both sides by σ:
2σ = 88 - μ + σ
Combine like terms:
2σ - σ = 88 - μ
σ = 88 - μ
Now we can substitute the expression for σ we obtained earlier:
σ = 88 - μ
-σ = 79 - μ
-9 = 79 - μ
μ = 79 + 9
μ = 88
Now we can substitute the value of μ into either of the original equations to solve for σ. Let's use the second equation for simplicity:
-1 = (79 - μ) / σ
-1 = (79 - 88) / σ
-1 = -9 / σ
Multiply both sides by σ:
-σ = -9
σ = 9
Therefore, the mean (μ) of the population is 88 and the standard deviation (σ) is 9.