QuestionFind the percentile rank for an IQ score of 100 using the standard normal distribution (z-distribution).

Studdy Solution

STEP 1

Assumptions1. The IQ score is normally distributed. . The mean IQ score is100.
3. The standard deviation of IQ scores is15.
4. We are looking for the percentile of an IQ score of100.

STEP 2

First, we need to standardize the IQ score to a z-score. The z-score is a measure of how many standard deviations an element is from the mean. It is calculated as $z = \frac{X - \mu}{\sigma}$ where- X is the value we are standardizing (in this case, the IQ score), - $\mu$ is the mean of the distribution, - $\sigma$ is the standard deviation of the distribution.

STEP 3

Now, plug in the given values for X, $\mu$ , and $\sigma$ to calculate the z-score.
$z = \frac{100 -100}{15}$

STEP 4

Calculate the z-score.
$z = \frac{100 -100}{15} =0$

STEP 5

The z-score of0 corresponds to the50th percentile in the standard normal distribution. This is because the standard normal distribution is symmetric about the mean, and the mean corresponds to the50th percentile.
Therefore, an IQ score of100 corresponds to the50th percentile.

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