(Normal Distribution) Suppose that the IQ of a randomly-selected student from a

Question

(Normal Distribution) Suppose that the IQ of a randomly-selected student from a university is normal, with a mean of 110 and a standard deviation of 20. (a) A quartile of a data set is one of four equally-sized intervals (e.g., the 1st quartile is the lowest 25% of the data, the 2nd is the next 25%, and so on), and the quartile midpoint is the mean of the upper and lower bounds of the quartile. What are the upper and lower bounds of the 3rd quartile and what is the midpoint of this quartile for the given distribution of IQ data? (b) Suppose we want to identify the middle 30% of the data, i.e., the interval whose midpoint is the mean, and includes 30% of the data. What are the upper and lower bounds of this interval?

Explanation / Answer

a) z-score corresponding to below 25% area = -0.67

So,

Corresponding lower bound of quartile = 110 - 0.67(20) = 96.6

Similarly,

Z is corresponding to above 20% area = 0.67

So,

Corresponding upper bound of the quartile = 110 + 0.67(20) = 123.4

Mean = Midpoint of quartiles = 110

b) Bound for Middle 30% of area can be found by calculating the score for which 35% area is to the left and 35% area is to the right.

z score corresponding to 35% area to its left = -0.39

Hence,

Lower bound = 110 - 0.39(20) = 104.2

Similarly,

Upper bound = 110 + 0.39(20) = 117.8

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