IQ is normally distributed with a mean of 100 and a standard deviation of 15
Suppose an individual is chosen at random:
a.) What is the probability the individual has an IQ greater than 122 ?
b.) MENSA is an organization whose members have IQs in the top $5 \%$.
What is the minimum IQ you would need to qualify for membership? (round to nearest whole number)
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Answer
Rounding to the nearest whole number, the minimum IQ to qualify for MENSA is \(\boxed{125}\).
Step 1 ：Convert the IQ score to a z-score using the formula \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the value we are looking to convert (in this case, 122), \(\mu\) is the mean (in this case, 100), and \(\sigma\) is the standard deviation (in this case, 15).
Step 2 ：Calculate the z-score for an IQ of 122: \(Z = \frac{122 - 100}{15} = \frac{22}{15} = 1.47\).
Step 3 ：Look up this z-score in a standard normal distribution table, or use a calculator, to find the probability that a randomly chosen individual has a z-score less than 1.47. This value is approximately 0.9292.
Step 4 ：Subtract the above probability from 1 to find the probability that the individual has an IQ greater than 122: \(P(X > 122) = 1 - P(X < 122) = 1 - 0.9292 = 0.0708\).
Step 5 ：\(\boxed{0.0708}\) or \(\boxed{7.08\%}\) is the probability that a randomly chosen individual has an IQ greater than 122.
Step 6 ：To find the minimum IQ you would need to qualify for MENSA, find the IQ score that corresponds to the top 5% of the distribution. This means finding the z-score that corresponds to a cumulative probability of 0.95 (since the top 5% is the complement of the bottom 95%).
Step 7 ：The z-score that corresponds to a cumulative probability of 0.95 is approximately 1.645.
Step 8 ：Convert this z-score back to an IQ score using the formula \(X = \mu + Z\sigma\), where \(\mu\) is the mean (in this case, 100), \(Z\) is the z-score (in this case, 1.645), and \(\sigma\) is the standard deviation (in this case, 15).
Step 9 ：Calculate the minimum IQ to qualify for MENSA: \(X = 100 + 1.645 \times 15 = 124.675\).
Step 10 ：Rounding to the nearest whole number, the minimum IQ to qualify for MENSA is \(\boxed{125}\).
