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Zorluk: OrtaNormal Distributions, Standard Deviation Curves, and Percentile Ranks

The distribution of scores on a graduate admissions examination is normally distributed with a mean of 540540 and a standard deviation of 3535. An applicant scored 610610 on this examination. If 800800 applicants scored higher than this applicant, which of the following is closest to the total number of applicants who took the examination?

  1. A
    8,0008,000
  2. B
    16,00016,000
  3. 32,00032,000Cevap
  4. D
    50,00050,000
  5. E
    64,00064,000

Cevap

32,00032,000
The z-score for a score of 610610 is calculated as z=(610−540)/35=2.0z = (610 - 540) / 35 = 2.0. By the 68-95-99.7 empirical rule for normal distributions, 95%95\% of all scores lie within 22 standard deviations of the mean. Because the distribution is symmetric, the remaining 5%5\% is split equally between the two tails, meaning 2.5%2.5\% of scores lie above z=2.0z = 2.0. Given that 800800 applicants scored higher than 610610, we set 0.025N=8000.025 N = 800, which gives total applicants N=32,000N = 32,000.

Adım Adım Çözüm

1
Calculate the z-score for a test score of 610610.
z=610−54035=7035=2.0z = \frac{610 - 540}{35} = \frac{70}{35} = 2.0
The z-score measures how many standard deviations the score is above the mean.
2
Determine the proportion of scores that lie above z=2.0z = 2.0 using the 68-95-99.7 empirical rule.
Proportion =100%−95%2=2.5%=0.025= \frac{100\% - 95\%}{2} = 2.5\% = 0.025
According to the empirical rule, 95%95\% of values lie within 22 standard deviations of the mean (between z=−2.0z = -2.0 and z=2.0z = 2.0). Due to symmetry, half of the remaining 5%5\%, or 2.5%2.5\%, lies strictly above z=2.0z = 2.0.
3
Set up an equation relating the number of higher-scoring applicants (800800) to the total number of applicants (NN).
0.025×N=800  ⟹  N=8000.025=32,0000.025 \times N = 800 \implies N = \frac{800}{0.025} = 32,000
Since 2.5%2.5\% of all applicants scored higher than 610610, dividing 800800 by 0.0250.025 yields the total population size.

Anahtar Kavram

Empirical Rule (68-95-99.7 Rule) and Standard Deviation Tail Areas
Tahmini Süre:1m 30s
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Konular

Normal Distributions, Standard Deviation Curves, and Percentile Ranks

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