Question

A normal distribution has μ = 32 and σ = 5.

(a) Find the z score corresponding to x = 27.

(b) Find the z score corresponding to x = 45.

(c) Find the raw score corresponding to z = −3.

(d) Find the raw score corresponding to z = 1.3.

Answer 1

We calculate z-scores by taking the value subtracted by the mean and dividing by the standard deviation. Conversely, we can find the raw score by multiplying the z-score with the standard deviation and adding the mean. This process enables comparison of scores from a normal distribution.

Step-by-step explanation:

The question is asking us to find various z-scores and raw scores corresponding to those z-scores in a normal distribution with mean (μ) of 32 and a standard deviation (σ) of 5.

1. To find the z-score for x = 27, use the formula z = (x - μ) / σ. Thus, z = (27 - 32) / 5 = -1.
2. To find the z-score for x = 45, use the same formula: z = (45 - 32) / 5 = 2.6.
3. To find the raw score corresponding to z = -3, rework the z-score formula to x = μ + (z ċ σ). So, x = 32 + (-3 ċ 5) = 32 - 15 = 17.
4. For z = 1.3, calculate the raw score: x = 32 + (1.3 ċ 5) = 32 + 6.5 = 38.5.

Through these calculations, we have translated raw scores into z-scores and back again, which allows comparing different scores from a normal distribution with their respective distances from the mean.