Consider the IQ model N(100, 16) one last time.
a) What IQ represents the 15th percentile?
b) What IQ represents the 98th percentile?
c) What’s the IQR of the IQs?
The Correct Answer and Explanation is :
The IQ model follows a normal distribution with a mean (( \mu )) of 100 and a standard deviation (( \sigma )) of 16, represented as ( N(100, 16) ).
a) What IQ represents the 15th percentile?
The 15th percentile corresponds to the point below which 15% of the data lies. Using a standard normal table (or ( Z )-score table), we find the ( Z )-score for 0.15:
[
Z_{15} \approx -1.04
]
Using the formula to convert ( Z )-scores to actual IQ values:
[
IQ = \mu + Z \cdot \sigma
]
[
IQ_{15} = 100 + (-1.04) \cdot 16 = 100 – 16.64 = 83.36
]
So, the IQ representing the 15th percentile is approximately 83.4.
b) What IQ represents the 98th percentile?
The 98th percentile corresponds to the point below which 98% of the data lies. Using a standard normal table:
[
Z_{98} \approx 2.05
]
Using the same formula:
[
IQ_{98} = 100 + (2.05) \cdot 16 = 100 + 32.8 = 132.8
]
So, the IQ representing the 98th percentile is approximately 132.8.
c) What’s the IQR of the IQs?
The interquartile range (IQR) is the difference between the 75th and 25th percentiles. From the ( Z )-table:
[
Z_{25} \approx -0.67, \quad Z_{75} \approx 0.67
]
Calculating the corresponding IQ values:
[
IQ_{25} = 100 + (-0.67) \cdot 16 = 100 – 10.72 = 89.28
]
[
IQ_{75} = 100 + (0.67) \cdot 16 = 100 + 10.72 = 110.72
]
The IQR is:
[
IQR = IQ_{75} – IQ_{25} = 110.72 – 89.28 = 21.44
]
So, the IQR of the IQs is approximately 21.4.
Explanation:
The normal distribution is symmetric, characterized by its mean and standard deviation. Percentiles correspond to specific ( Z )-scores, which are standard deviations away from the mean. By converting ( Z )-scores using ( Z = \frac{X – \mu}{\sigma} ), we find specific IQ values for given percentiles. The IQR, representing the spread of the middle 50% of values, is particularly useful for understanding variability. This approach ensures precision and highlights the statistical properties of the normal distribution.