Question

If the reliability coefficient of an instrument is .91, and the standard deviation is 10, then a client's score of 59 could be interpreted that 95 percent of the time his/her true score will fall between _____ and _____ using standard error of measurement​

Answer 1

To interpret the client’s true score with 95% confidence using a reliability coefficient of .91 and SD of 10, calculate the SEM and then find the range using the z-score for 95% confidence. The client's true score falls between 53.12 and 64.88.

Step-by-step explanation:

To calculate the range in which a client’s true score will likely fall with 95% confidence using the standard error of measurement, we first need to calculate the standard error of measurement (SEM) using the given reliability coefficient.

The formula for SEM is:

​SEM = SD √(1 − rxx)

Where:

• SD is the standard deviation.
• rxx is the reliability coefficient of the instrument.

Plugging in the values, we get:

SEM = 10 √(1 - 0.91) = 10 √(0.09) = 10 √(0.3) = 3

To find the range with 95% confidence, we need to use the z-score associated with 95% confidence which is approximately 1.96. Thus, the range is calculated as:

Range = Client's score ± (z * SEM)

Range = 59 ± (1.96 * 3)

This yields the range:

59 - (1.96 * 3) = 59 - 5.88 = 53.12

59 + (1.96 * 3) = 59 + 5.88 = 64.88

A client's true score will likely fall between 53.12 and 64.88, 95% of the time.

This demonstrates the 95 percent confidence level, which means that if you were to take 100 different samples and compute a confidence interval for each sample, approximately 95 percent of the confidence intervals would contain the true mean.