Adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15. a. What IQ score represents the 95th percentile? b. What IQ score represents the 50th percentile? Show how you got the answer step by step, clearly just trying to check my work. Thanks!
The Correct Answer and Explanation is:
Let’s solve this step-by-step using the standard normal distribution.
Given:
- Mean (μ) = 100
- Standard deviation (σ) = 15
- Distribution: Normal
Part a: What IQ score represents the 95th percentile?
This means we want the IQ score such that 95% of the population scores below it. This corresponds to a z-score at the 95th percentile.
Step 1: Find the z-score for the 95th percentile
Using a standard normal table or calculator:z0.95≈1.645z_{0.95} \approx 1.645z0.95≈1.645
Step 2: Use the z-score formula to find the IQ score:
X=μ+zσX = \mu + z\sigmaX=μ+zσX=100+(1.645)(15)=100+24.675=124.675X = 100 + (1.645)(15) = 100 + 24.675 = 124.675X=100+(1.645)(15)=100+24.675=124.675Rounded to the nearest whole number: 125\text{Rounded to the nearest whole number: } \boxed{125}Rounded to the nearest whole number: 125
Part b: What IQ score represents the 50th percentile?
The 50th percentile is the median, which for a normal distribution is the mean.
So,100\boxed{100}100
Explanation
IQ scores follow a bell-shaped curve known as the normal distribution. This type of distribution is symmetric around its mean. The standard deviation tells us how spread out the values are around the mean. In this case, IQ scores have a mean of 100 and a standard deviation of 15.
Percentiles help us understand how a given score compares to others. The 95th percentile is a score that is higher than 95% of all other scores. To find this, we use the z-score, which tells us how many standard deviations a value is from the mean. For the 95th percentile, the z-score is approximately 1.645. We then convert this z-score back into an actual IQ score using the formula:X=μ+zσX = \mu + z\sigmaX=μ+zσ
Plugging in the values gives an IQ score of around 125. This means an IQ of 125 is higher than 95% of adult IQ scores.
The 50th percentile, or the median, is simply the center of the distribution. Since the normal distribution is symmetric, the mean and median are the same. So the IQ score that represents the 50th percentile is 100, the average.
These calculations are useful in understanding standardized test results, cognitive assessments, and comparisons across populations.
