Question

On a planet far far away from Earth, IQ of the ruling species is normally distributed with a mean of 108 and a standard deviation of 17. Suppose one individual is randomly chosen. Let X = IQ of an individual.

Answer 1

Explanation:

Find the probability that X is less than 92:

We can use the cumulative distribution function (CDF) of a normal distribution to find the probability that X is less than 92. The CDF of a normal distribution with mean μ and standard deviation σ is given by:

F(x) = (1/2) * [1 + erf((x-μ)/(σ * sqrt(2)))]

Where erf is the error function.

So, the probability that X is less than 92 can be calculated as:

P(X < 92) = F(92) = (1/2) * [1 + erf((92-108)/(17 * sqrt(2)))]

This value can be calculated using a calculator or statistical software.

Find the probability that X is between 92 and 126:

We can use the CDF to find the probability that X is between 92 and 126. The probability that X is between 92 and 126 is equal to the difference between the CDF values at 126 and 92:

P(92 < X < 126) = F(126) - F(92)

This value can also be calculated using a calculator or statistical software.

Find the IQ of an individual that is in the top 10% of the population:

To find the IQ of an individual that is in the top 10% of the population, we need to find the value of X that corresponds to a cumulative probability of 0.9. This can be done by solving for X in the CDF equation:

F(X) = 0.9

X = μ + σ * inverse_CDF(0.9)

Where inverse_CDF is the inverse cumulative distribution function and can be calculated using a calculator or statistical software. The value of X that corresponds to a cumulative probability of 0.9 is the IQ of an individual that is in the top 10% of the population.