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Amanda
2 months ago
11

Seventy new jobs are opening up at an automobile manufacturing plant, and 1000 applicants show up for the 70 positions. To selec

t the best 70 from among the applicants, the company gives a test that covers mechanical skill, manual dexterity, and mathematical ability. The mean grade on this test turns out to be 60, and the scores have a standard deviation of 6. Can a person who scores 84 count on getting one of the jobs? [Hint: Use Chebyshev’s theorem.] Assume that the distribution is symmetric about the mean
Business
1 answer:
Free_Kalibri [3.7K]2 months ago
3 0

Answer:

  • As explained below, with the individual’s score in the 0.03125 fraction of top candidates, they can anticipate securing a position.

Explanation:

Utilizing Chebyshev’s Theorem is key.

This theorem is valid for any dataset, irrespective of its shape.

Chebyshev's Theorem states that at least 1−1/k² of the data falls within k standard deviations from the mean.

For this data set, the specifics are:

  • mean: 60
  • standard deviation: 6
  • score: 84

The number of standard deviations that 84 is from the mean can be calculated as:

  • k = (score - mean) / standard deviation
  • k = (84 - 60) / 6 = 24 / 6 = 4

Hence, the individual’s score is 4 standard deviations above the mean.

How significant is this?

According to Chebyshev’s Theorem, at least 1−1/k² of the data is within k standard deviations from the mean. Setting k = 4 gives us:

  • 1 - 1/4² = 1 - 1/16 = 0.9375

  • This implies that half of 1 - 0.9375 exceed k = 4: 0.03125

  • Consequently, 1 - 0.03125 is below k = 4: 0.96875

With 70 job openings and 1,000 applicants, the ratio is 70/1,000 = 0.07, indicating the company seeks the top 0.07 of applicants.

Given the individual scores in the top 0.03125 of applicants, they can expect to obtain a job.

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