Question

Nationwide, sixth graders took a test of reading comprehension. Sam’s z-score for this test is –1.47. Write a note to his parents that explains what his z-score means.

a) His score is below the mean.
b) His score is 1.47 standard deviations below the mean.
c) His score is above the mean.
d) His score is 1.47 standard deviations above the mean.

Answer 1

z-score means his score is 1.47 standard deviations below the mean. Thus, the correct answer is option b) His score is 1.47 standard deviations below the mean.

Step-by-step explanation:

Sam's z-score of -1.47 indicates that his performance on the sixth-grade reading comprehension test is 1.47 standard deviations below the mean. The z-score is a measure of how many standard deviations a data point is from the mean of a distribution. In this case, a negative z-score suggests that Sam's score is below the average performance of the sixth-grade population.

To calculate the z-score, we use the formula:

`\[Z = \frac{{X - \mu}}{{\sigma}}\]`

where Z is the z-score, X is Sam's score,
`\(\mu\)` is the mean, and
`\(\sigma\)` is the standard deviation. Since Sam's z-score is negative, we know that his score is below the mean. The magnitude of the z-score, 1.47, indicates that Sam's score is 1.47 standard deviations below the mean.

Understanding z-scores is crucial for interpreting how an individual's score compares to the overall performance of a group. In Sam's case, being 1.47 standard deviations below the mean suggests that his reading comprehension performance is lower than the majority of sixth-grade students who took the test nationwide. Parents can use this information to gain insights into Sam's relative standing and, if necessary, consider additional support or resources to help improve his reading comprehension skills. Therefore, the correct answer is option b) His score is 1.47 standard deviations below the mean.