Final answer:
A z-score measures how many standard deviations a score is from the mean. It uses the formula X - μ / σ. Z-scores of 1, -1, 1.5, and corresponding raw scores of 90 and 50 for z-scores of 4 and -4, respectively, were calculated.
Step-by-step explanation:
The z-score is a measure of how many standard deviations a data point is from the mean of a data set.
Calculating Z-Scores
To calculate the z-score for a particular score, use the formula: z = (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation.
a The z-score for 75 is 1. 75 - 70 / 5 = 1.
b The z-score for 65 is -1. 65 - 70 / 5 = -1.
c The z-score for 77.5 is 1.5. 77.5 - 70 / 5 = 1.5.
d A raw score with a z-score of 4 would be 90. 70 + 4 * 5 = 90.
e A raw score with a z-score of -4 would be 50. 70 + -4 * 5 = 50.
Examples based on z-scores:
f A raw score between 0 and 1 z-scores could be 72, as it is slightly above the mean.
g A raw score between -1 and 0 z-scores could be 68, as it is slightly below the mean.
Interpreting Z-Scores
High z-scores indicate scores far above the mean, while low (including negative) z-scores indicate scores below the mean. The higher or lower the z-score, the more unusual the result is in the context of the data set.