Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10, what is the probability that a person who takes the test will score between 90 and 110?

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Scores on an IQ test are normally distributed. If the test has a mean of 100 and a SD of 10, what is the probability that a person who takes the test will score between 90 and 110?

What is an IQ test

To find the probability that a person who takes the test will score between 90 and 110, we can use the properties of the normal distribution.

1. Calculate the z-scores for the lower and upper bounds:
The z-score measures how many standard deviations a value is from the mean. The formula to calculate the z-score is:
z = (x - μ) / σ

For the lower bound, x = 90, μ = 100, and σ = 10:
z1 = (90 - 100) / 10 = -1

For the upper bound, x = 110, μ = 100, and σ = 10:
z2 = (110 - 100) / 10 = 1

2. Look up the z-scores in the standard normal distribution table:
The standard normal distribution table provides the area under the curve for a given z-score.

The area to the left of -1 is 0.1587, and the area to the left of 1 is 0.8413.

3. Calculate the probability by subtracting the area to the left of the lower bound from the area to the left of the upper bound:
P(90 ≤ X ≤ 110) = P(X ≤ 110) - P(X ≤ 90)
= 0.8413 - 0.1587
= 0.6826

Therefore, the probability that a person who takes the IQ test will score between 90 and 110 is approximately 0.6826, or 68.26%.