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Scores on the Wechsler Adult Intelligence Scale (an IQ test) for the 20- to 34-year-old age group are approximately Normally distributed with μ=110 and σ=25. What percent of people aged 20 to 34 have IQs between 125 and 150?

Answer :

To find the percentage of people aged 20 to 34 with IQs between 125 and 150, when the IQ scores are normally distributed with a mean ([tex]\mu[/tex]) of 110 and a standard deviation ([tex]\sigma[/tex]) of 25, we will use the properties of the normal distribution.

Step 1: Standardize the Scores

We will convert the IQ scores to z-scores using the formula:

[tex]z = \frac{X - \mu}{\sigma}[/tex]

  • For an IQ of 125:
    [tex]z = \frac{125 - 110}{25} = \frac{15}{25} = 0.6[/tex]

  • For an IQ of 150:
    [tex]z = \frac{150 - 110}{25} = \frac{40}{25} = 1.6[/tex]

Step 2: Find the Corresponding Percentages

Using a standard normal distribution table (z-table), or a calculator with normal distribution functions, we find the area (or probability) that corresponds to these z-scores:

  • The probability (or area) for [tex]z = 0.6[/tex] is approximately [tex]0.7257[/tex].

  • The probability (or area) for [tex]z = 1.6[/tex] is approximately [tex]0.9452[/tex].

Step 3: Calculate the Percentage

To find the percentage of people who have IQs between 125 and 150, we subtract the smaller area from the larger area:

[tex]0.9452 - 0.7257 = 0.2195[/tex]

This means approximately [tex]21.95\%[/tex] of people aged 20 to 34 have IQs between 125 and 150.

Therefore, according to our calculations based on a normally distributed data assumption, about [tex]21.95\%[/tex] of individuals in this age group would have IQ scores in the specified range.

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