Question

Suppose the scores on an IQ test are approximately normally distributed with a mean of 100 and a standard deviation of 10. What is the probability that a randomly selected person scores above 125 on the IQ test

Answer 1

Answer:

The probability that a randomly selected person scores above 125 on the IQ test is 0.0062.

Step-by-step explanation:

We are given that the the scores on an IQ test are approximately normally distributed with a mean of 100 and a standard deviation of 10.

Let X = scores on an IQ test

The z-score probability distribution is given by ;

                Z = $\frac{ X-\mu}{\sigma}} }$ ~ N(0,1)

where, $\mu$ = mean score = 100

            $\sigma$ = standard deviation = 10

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

Now, the probability that a randomly selected person scores above 125 on the IQ test is given by = P(X > 125)

       P(X > 125) = P( $\frac{ X-\mu}{\sigma}} }$ > $\frac{ 125-100}{10}} }$ ) = P(Z > 2.50) = 1 - P(Z $\leq$ 2.50)

                                                        = 1 - 0.9938 = 0.0062                                The above probability is calculated using z table by looking at value of x = 2.50 in the z table which have an area of 0.99379.

Therefore, probability that a randomly selected person scores above 125 on the IQ test is 0.0062.

Answer 2

Answer:

0.62% probability that a randomly selected person scores above 125 on the IQ test

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 100, \sigma = 10$

What is the probability that a randomly selected person scores above 125 on the IQ test

This is 1 subtracted by the pvalue of Z when X = 125. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{125 - 100}{10}$

$Z = 2.5$

$Z = 2.5$ has a pvalue of 0.9938

1 - 0.9938 = 0.0062

0.62% probability that a randomly selected person scores above 125 on the IQ test