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Chapter 6: Problem 51

Critical reading SAT scores are distributed as \(N(500,100)\) a. Find the SAT score at the 75 th percentile. b. Find the SAT score at the 25 th percentile. c. Find the interquartile range for SAT scores. d. Is the interquartile range larger or smaller than the standard deviation? Explain.

Short Answer

Expert verified
a. The SAT score at the 75th percentile is 567.4. \n b. The SAT score at the 25th percentile is 432.6. \n c. The interquartile range for SAT scores is 134.8. \n d. The interquartile range is larger than the standard deviation.

Step by step solution

01

Calculation of the 75th percentile score

The score corresponding to any given percentile can be calculated by using the formula: Percentile Score = mean + z * standard deviationwhere z is the z-value corresponding to the percentile. For the 75th percentile, the z value is 0.674 (this is a known value based on z-table or can be calculated using z-score calculator). So, the 75th percentile score would be 75th Percentile = \(500 + 0.674*100 = 567.4\)
02

Calculation of the 25th percentile score

Following the same process as above, for the 25th percentile, the z value is -0.674. Therefore, the 25th percentile score will be: 25th Percentile = \(500 + (-0.674)*100 = 432.6\)
03

Calculation of interquartile range (IQR)

The interquartile range is the difference between the 75th percentile score and the 25th percentile score. Therefore, IQR = \(75th percentile score - 25th percentile score = 567.4 - 432.6 = 134.8\)
04

Compare IQR with standard deviation

We know the standard deviation (\( \sigma \)) from the given data is 100. Now, as per the calculated values, \( IQR = 134.8 \). So, the interquartile range is larger than the standard deviation.

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