Question

A person must score in the upper 8% of the population on an IQ test to qualify for membership in MENSA, the international high-IQ society. IQ scores are normally distributed with a mean of 115 and a standard deviation of 17. (a). [7pts] What IQ score must a person get to qualify

Answer 1

Answer: The person must score an IQ score of 139 to qualify.

Explanation:

Since the IQ scores are normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = scores on the test.

µ = mean score

σ = standard deviation

From the information given,

µ = 115

σ = 17

The probability value for the upper 8% of the population would be (1 - 8/100) = (1 - 0.08) = 0.92

Looking at the normal distribution table, the z score corresponding to the probability value is 1.41

Therefore,

1.41 = (x - 115)/17

Cross multiplying by 17, it becomes

1.41 × 17 = x - 115

23.97 = x - 115

x = 23.97 + 115

x = 139 to the nearest whole number

Answer 2

A person must get an IQ score of at least 138.885 to qualify.

Explanation:

Problems of normally distributed samples are 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 = (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 = 115, \sigma = 17`

(a). [7pts] What IQ score must a person get to qualify

Top 8%, so at least the 100-8 = 92th percentile.

Scores of X and higher, in which X is found when Z has a pvalue of 0.92. So X when Z = 1.405.

`Z = (X - \mu)/(\sigma)`

`1.405 = (X - 115)/(17)`

`X - 115 = 17*1.405`

`X = 138.885`

A person must get an IQ score of at least 138.885 to qualify.