fraction of the IQ scores would be between 97 and 126? 2. What is the 95th percentile of this normal distribution?
The distribution of IQ scores for high school graduates is normally distributed with u=104 and o=16. 1. What fraction of the IQ scores would be between 97 and 126? 2. What is the 95th percentile of this normal distribution?
The Correct Answer and Explanation is:
We are given that IQ scores are normally distributed with a mean (μ) of 104 and a standard deviation (σ) of 16. We will use the properties of the normal distribution to answer both questions.
1. What fraction of the IQ scores would be between 97 and 126?
To find this, we first convert the raw scores to z-scores using the formula:z=X−μσz = \frac{X – \mu}{\sigma}z=σX−μ
For X = 97:z=97−10416=−716=−0.4375z = \frac{97 – 104}{16} = \frac{-7}{16} = -0.4375z=1697−104=16−7=−0.4375
For X = 126:z=126−10416=2216=1.375z = \frac{126 – 104}{16} = \frac{22}{16} = 1.375z=16126−104=1622=1.375
Now, using the standard normal distribution table or a calculator:
- The area to the left of z = -0.44 is approximately 0.3300
- The area to the left of z = 1.38 is approximately 0.9162
To find the fraction between these z-values, subtract the lower area from the higher:0.9162−0.3300=0.58620.9162 – 0.3300 = 0.58620.9162−0.3300=0.5862
So, about 58.6 percent of IQ scores fall between 97 and 126.
2. What is the 95th percentile of this normal distribution?
The 95th percentile corresponds to a z-score where 95 percent of the data lies to the left of that value. Using a standard normal table or calculator:
- The z-score for the 95th percentile is approximately 1.645
Now convert this z-score back to an IQ score using:X=μ+zσ=104+(1.645)(16)=104+26.32=130.32X = \mu + z\sigma = 104 + (1.645)(16) = 104 + 26.32 = 130.32X=μ+zσ=104+(1.645)(16)=104+26.32=130.32
Thus, the 95th percentile of this distribution is approximately 130.3.
Summary
- About 58.6% of IQ scores lie between 97 and 126.
- The 95th percentile IQ score is approximately 130.3.
This analysis uses the properties of the normal distribution to translate between raw scores and z-scores, allowing us to determine areas under the curve and specific percentiles.
