Questions: A set of data values is normally distributed with a mean of 60 and a standard deviation of 8. Give the standard score and approximate percentile for a data value of 58. The standard score for 58 is z=

A set of data values is normally distributed with a mean of 60 and a standard deviation of 8. Give the standard score and approximate percentile for a data value of 58.

The standard score for 58 is z=

Transcript text: A set of data values is normally distributed with a mean of 60 and a standard deviation of 8. Give the standard score and approximate percentile for a data value of 58. The standard score for 58 is $z=$ $\square$ (Do not round until the final answer. Then round to the nearest hundredth as needed.)

Solution

Solution Steps

To find the standard score (z-score) for a data value in a normally distributed set, we use the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where $ X $ is the data value, $ \mu $ is the mean, and $ \sigma $ is the standard deviation. Once we have the z-score, we can use the provided table to find the approximate percentile.

Step 1: Calculate the z-score

To find the standard score (z-score) for a data value in a normally distributed set, we use the formula: \[ z = \frac{X - \mu}{\sigma} \] where:

$ X = 58 $ (data value)
$ \mu = 60 $ (mean)
$ \sigma = 8 $ (standard deviation)

Substituting the values, we get: \[ z = \frac{58 - 60}{8} = \frac{-2}{8} = -0.25 \]

Step 2: Determine the approximate percentile

Using the provided table of standard scores and percentiles, we find the percentile corresponding to $ z = -0.25 $. According to the table, the percentile for $ z = -0.25 $ is approximately $ 40.13 $.