Question

The grades on a statistics mid-term for a high school are normally distributed, with µ = 81 and σ = 6.3. Calculate the z-scores for each of the following exam grades. Draw and label a sketch for each example. 65, 83, 93, 100

Answer 1

Scores' z-values: 65 (Z ≈ -2.54), 83 (Z ≈ 0.32), 93 (Z ≈ 1.90), 100 (Z ≈ 3.02), visualized on a standard normal curve.

To calculate the z-scores for each exam grade given the mean (\( \mu = 81 \)) and standard deviation sigma = 6.3, we'll use the formula for the z-score:

`\[ Z = (X - \mu)/(\sigma) \]`

Where:

X is the individual score.

mu is the mean.

sigma is the standard deviation.

Let's calculate the z-scores for each exam grade:

1. X = 65:

`\[ Z = (65 - 81)/(6.3) = (-16)/(6.3) \approx -2.54 \]`

2. X = 83:

`\[ Z = (83 - 81)/(6.3) = (2)/(6.3) \approx 0.32 \]`

3. X = 93:

`\[ Z = (93 - 81)/(6.3) = (12)/(6.3) \approx 1.90 \]`

4. X = 100:

`\[ Z = (100 - 81)/(6.3) = (19)/(6.3) \approx 3.02 \]`

Now, let's sketch and label each of these z-scores on a standard normal distribution curve. The z-scores represent the number of standard deviations away from the mean:

Z = -2.54: This value lies approximately 2.54 standard deviations below the mean.

Z = 0.32: This value lies approximately 0.32 standard deviations above the mean.

Z = 1.90: This value lies approximately 1.90 standard deviations above the mean.

Z = 3.02: This value lies approximately 3.02 standard deviations above the mean.

Understanding the z-score helps identify where each individual exam grade falls within the distribution in terms of standard deviations from the mean.