Question

source: new york times almanac.

1. labor charges the average labor charge for automobile mechanics is $54 per hour. the standard deviation is $4. find the minimum percentage of data values that will fall within the range of $48 to $60. use chebyshevs theorem.

Explanation:

Step1: Calculate the number of standard deviations away from the mean

The mean $\mu = 54$, and we want to find $k$ for the range.
For the lower bound: $54 - k\times4=48$, so $k\times4 = 54 - 48=6$, then $k=\frac{6}{4}=1.5$.
For the upper bound: $54 + k\times4=60$, so $k\times4=60 - 54 = 6$, then $k = 1.5$.

Step2: Apply Chebyshev's theorem

Chebyshev's theorem states that the proportion of data within $k$ standard deviations of the mean is at least $1-\frac{1}{k^{2}}$.
Substitute $k = 1.5$ into the formula: $1-\frac{1}{1.5^{2}}=1-\frac{1}{2.25}=\frac{2.25 - 1}{2.25}=\frac{1.25}{2.25}=\frac{5}{9}\approx0.556$.

Answer:

The minimum percentage of data values that fall within the range is approximately $55.6\%$.