Question

Find the indicated IQ score. The graph depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard
deviation of 15 (as on the Wechsler test).
The shaded area under the curve is 0.5675.

Answer 1

Answer:

IQ score of 102.55

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, which is also the shaded area under the curve, 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:

$\mu = 100, \sigma = 15$

The shaded area under the curve is 0.5675.

This means that Z has a pvalue of 0.5675.

So, we have to find X when Z = 0.17.

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

$0.17 = \frac{X - 100}{15}$

$X - 100 = 15*0.17$

$X = 102.55$

IQ score of 102.55

Answer 2

Using the Zscore formula, the indicated IQ score of the normally distributed data is 102.25

• Using the normal distribution table ; 0.5675 has a Zscore which corresponds to 0.17

Recall :

Zscore = (Score - Mean) / standard deviation

0.17 = (Score - 100) / 15

15 × 0.17 = Score - 100

2.25 = score - 100

Score = 100 + 2.25

Score = 102.25

Therefore, the indicated IQ score is 102.25

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