The area of the shaded region is approximately 31.74%. This represents the 2.5% of the population that has IQ scores below 85 or above 115.
Since the IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
we know that 68% of the population will have IQ scores within 15 points of the mean (between 85 and 115), 95% of the population will have IQ scores within 2 standard deviations of the mean (between 70 and 130), and 99.7% of the population will have IQ scores within 3 standard deviations of the mean (between 55 and 145).
The shaded region on the graph represents the 2.5% of the population that has IQ scores below 85 or above 115.
Since the normal distribution is symmetrical, the shaded area on the left side of the graph (IQ scores below 85) is equal to the shaded area on the right side of the graph (IQ scores above 115).
Therefore, to find the area of the shaded region, we need to find the area of the shaded region on either side of the graph.
We can do this by calculating the area between the mean (100) and 85, and multiplying that area by 2.
The area between the mean and 85 can be calculated using the formula for the area under the standard normal curve:
A = 0.5 * (1 - erf((x - μ) / σ))
where:
A is the area
erf is the error function
x is the IQ score (85 in this case)
μ is the mean (100 in this case)
σ is the standard deviation (15 in this case)
Plugging in the values, we get:
A = 0.5 * (1 - erf((85 - 100) / 15))
A ≈ 0.1587
Therefore, the area of the shaded region is approximately 0.1587 * 2 = 0.3174, or 31.74%.
Question
Find the area of the shaded region. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. The shade region is 125.