Questions: Assume that adults have IQ scores that are normally distributed with a mean of μ=105 and a standard deviation σ=20. Find the probability that a randomly selected adult has an IQ between 95 and 115. The probability that a randomly selected adult has an IQ between 95 and 115 is

Solution Steps

We are given that the IQ scores of adults are normally distributed with a mean \( \mu = 105 \) and a standard deviation \( \sigma = 20 \).

To find the probability that a randomly selected adult has an IQ between 95 and 115, we first calculate the Z-scores for the lower and upper bounds of the range.

The Z-score is calculated using the formula: \[ Z = \frac{X - \mu}{\sigma} \] For the lower bound \( X = 95 \): \[ Z_{start} = \frac{95 - 105}{20} = -0.5 \]

For the upper bound \( X = 115 \): \[ Z_{end} = \frac{115 - 105}{20} = 0.5 \]

The probability that a randomly selected adult has an IQ between 95 and 115 is given by: \[ P(95 < X < 115) = \Phi(Z_{end}) - \Phi(Z_{start}) \] Where \( \Phi \) is the cumulative distribution function (CDF) of the standard normal distribution.

Using the Z-scores calculated: \[ P(95 < X < 115) = \Phi(0.5) - \Phi(-0.5) \]

From the output, we find: \[ P(95 < X < 115) = 0.3829 \]

Final Answer