Adult IQ scores are normally distributed with mean 100 and standard deviation 15. Find the IQ score that separates the adults with the highest 3% of IQ scores from the rest of the adults. Round your answer to the nearest whole number.
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To find the IQ score that separates the adults with the highest 3% of IQ scores from the rest of the adults, we can use the concept of z-scores.
First, we need to find the z-score that corresponds to the top 3% of the normal distribution. This can be done using the inverse normal distribution function (also known as the inverse cumulative distribution function). Specifically, we want to find the z-score corresponding to a cumulative probability of 0.97 (since the top 3% equals 100% - 97%).
Using a standard normal distribution table or a statistical calculator, we can find that the z-score corresponding to a cumulative probability of 0.97 is approximately 1.88.
Next, we can use the formula for z-scores to find the corresponding IQ score:
z = (x - μ) / σ
where:
z = z-score
x = IQ score we want to find
μ = mean of the distribution (100)
σ = standard deviation of the distribution (15)
Rearranging the formula to solve for x:
x = z * σ + μ
Substituting the known values:
x = 1.88 * 15 + 100
Calculating the expression:
x ≈ 28.2 + 100
x ≈ 128.2
Rounding the answer to the nearest whole number:
x ≈ 128
Therefore, the IQ score that separates the adults with the highest 3% of IQ scores from the rest of the adults is approximately 128.