In a population of exam scores, a score of X = 48 corresponds to z = +1.00 and a score of X = 36 corresponds to z = -0.50. Find the mean and standard deviation of the population.

To find the mean and standard deviation, we need to use the formula for z-score:

z = (X - μ) / σ

Where:
X = individual score
μ = population mean
σ = population standard deviation

Given that a score of X = 48 corresponds to z = +1.00 and a score of X = 36 corresponds to z = -0.50, we can set up two equations:

1.00 = (48 - μ) / σ ---(1)
-0.50 = (36 - μ) / σ ---(2)

To solve this system of equations, we can use the method of substitution. Let's solve equation (2) for μ:

-0.50σ = 36 - μ
μ = 36 + 0.50σ

Now substitute this value of μ in equation (1):

1.00 = (48 - (36 + 0.50σ)) / σ
1.00 = (12 - 0.50σ) / σ

Cross-multiply to get rid of the fraction:

1.00σ = 12 - 0.50σ

1.00σ + 0.50σ = 12

1.50σ = 12

σ = 12 / 1.50

σ = 8

Now substitute this value of σ in equation (2) to solve for μ:

-0.50 = (36 - μ) / 8

-4 = 36 - μ

μ = 36 + 4

μ = 40

Therefore, the mean (μ) of the population is 40 and the standard deviation (σ) is 8.

To find the mean and standard deviation of the population, you need to use the formula for calculating z-scores.

The formula for calculating the z-score is:

z = (X - mean) / standard deviation

Since we have two data points, we can set up two equations using the given information:

Equation 1: 1 = (48 - mean) / standard deviation
Equation 2: -0.5 = (36 - mean) / standard deviation

We can solve this system of equations to find the mean and standard deviation.

First, let's isolate mean in Equation 1:
48 - mean = standard deviation

Now, substitute this into Equation 2:
-0.5 = (36 - (48 - mean)) / standard deviation

Simplify:
-0.5 = (36 - 48 + mean) / standard deviation
-0.5 = (mean - 12) / standard deviation

Now, multiply both sides of the equation by standard deviation:
-0.5 * standard deviation = mean - 12

Rearrange the equation:
mean = -0.5 * standard deviation + 12

Substitute this value for mean into Equation 1:
1 = (48 - (-0.5 * standard deviation + 12)) / standard deviation

Simplify:
1 = (48 + 0.5 * standard deviation - 12) / standard deviation
1 = (36 + 0.5 * standard deviation) / standard deviation

Multiply both sides of the equation by standard deviation:
standard deviation = 36 + 0.5 * standard deviation

Now, subtract 0.5 * standard deviation from both sides:
standard deviation - 0.5 * standard deviation = 36

Combine like terms:
0.5 * standard deviation = 36

Divide both sides by 0.5:
standard deviation = 72

Now, substitute this value for standard deviation into the equation for mean:
mean = -0.5 * 72 + 12

Simplify:
mean = -36 + 12
mean = -24

Therefore, the mean of the population is -24 and the standard deviation is 72.

The difference between the 48 and the 36 is 1.5 standard deviations. Thus

1.5 sigma = 12,
and sigma (the standard deviation) = 8

48 is one sigma above the mean, so the mean is 40.