Questions: An intelligence scale for children is approximately normally distributed, with mean 100 and standard deviation 15. Complete parts (a) through (f) below. (a) What is the probability that a randomly selected test taker will score above 125 ? 0.0478 (b) What is the probability that a randomly selected test taker will score below 90 ? 0.2525 (c) What proportion of test takers will score between 110 and 140 ? 0.2487 (d) Would it be unusual for a randomly selected child to have a score above 150 ? because P(X>150)= .

Solution Steps

To find the probability that a randomly selected test taker will score above \( 125 \), we first calculate the cumulative distribution function (CDF) at \( x = 125 \):

\[ P(X \leq 125) = CDF(125) \approx 0.9522 \]

Thus, the probability of scoring above \( 125 \) is given by:

\[ P(X > 125) = 1 - P(X \leq 125) = 1 - 0.9522 \approx 0.0478 \]

Next, we calculate the probability that a randomly selected test taker will score below \( 90 \):

\[ P(X \leq 90) = CDF(90) \approx 0.2525 \]

This means that the probability of scoring below \( 90 \) is:

\[ P(X < 90) \approx 0.2525 \]

To find the proportion of test takers that will score between \( 110 \) and \( 140 \), we calculate the CDF at both values:

\[ P(X \leq 110) = CDF(110) \approx 0.5000 \] \[ P(X \leq 140) = CDF(140) \approx 0.7487 \]

The proportion of test takers scoring between \( 110 \) and \( 140 \) is:

\[ P(110 < X < 140) = P(X \leq 140) - P(X \leq 110) \approx 0.7487 - 0.5000 = 0.2487 \]

Final Answer