Incomes of workers in an automobile company in Michigan are known to be right - skewed with a mean equal to $36,100. If at least 8/9 of all incomes are in the range of $29,600 to $42,800, and this was based on Tchebysheff's Theorem, what is the standard deviation for the auto workers?
A) $6,500
B) $6,700
C) Approximately $4,755
D) $2,200
E) $2,600

Question

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According to Tchebysheff's Theorem, at least 1 - 1/k^2 of all data values fall within k standard deviations from the mean. In this case, we want at least 8/9 of all incomes to be in the range of $29,600 to $42,800.

First, we need to find the range of values that is k standard deviations away from the mean and contains at least 8/9 of the data. We can write this as:

$29,600 ≤ X ≤ $42,800

Subtracting the mean from both sides, we get:

-$6,500 ≤ X - $36,100 ≤ $6,700

Next, we need to find the value of k such that at least 8/9 of the data falls within this range. Using Tchebysheff's Theorem, we know that:

1 - 1/k^2 ≥ 8/9

Solving for k gives:

k ≥ sqrt(9/8) ≈ 1.06

So we need at least 1.06 standard deviations to capture at least 8/9 of the data. Therefore, we can solve for the standard deviation by setting k = 1.06:

$6,700 = 1.06σ

Solving for σ gives:

σ ≈ $6,320/1.06 ≈ $5,962

The closest answer choice is D) $2,200. However, this is not the correct answer. We made an error in our calculation of k. The correct calculation for k is:

k ≥ sqrt(9/8) ≈ 1.03

So we need at least 1.03 standard deviations to capture at least 8/9 of the data. Therefore, we can solve for the standard deviation by setting k = 1.03:

$6,700 = 1.03σ

Solving for σ gives:

σ ≈ $6,504

The closest answer choice to this value is A) $6,500. Therefore, the answer is:

Answer: A
 We used Tchebysheff's Theorem to find that we need at least 1.03 standard deviations to capture at least 8/9 of the data. Solving for the standard deviation gave us a value of $6,504, which is closest to answer choice A) $6,500.

Incomes of Workers in an Automobile Company in Michigan Are