Question

1. costs to train employees for a certain type of job, it costs a company an average of $231 to train an employee to perform the task. the standard deviation is $5. find the minimum percentage of data values that will fall in the range of $219 to $243. use chebyshevs theorem.
2. delivery charges the average delivery charge for a refrigerator is $32. the standard deviation is $4. find the minimum percentage of data values that will fall in the range of $20 to $44. use chebyshevs theorem.

Explanation:

Step1: Calculate the number of standard deviations ($k$)

For problem 18:
The mean $\mu = 231$, standard deviation $\sigma=5$.
For the lower bound: $231 - k\times5=219$, so $k=\frac{231 - 219}{5}=\frac{12}{5} = 2.4$.
For the upper bound: $231 + k\times5=243$, so $k=\frac{243 - 231}{5}=\frac{12}{5}=2.4$.
For problem 19:
The mean $\mu = 32$, standard deviation $\sigma = 4$.
For the lower bound: $32-k\times4=20$, so $k=\frac{32 - 20}{4}=\frac{12}{4}=3$.
For the upper bound: $32 + k\times4=44$, so $k=\frac{44 - 32}{4}=\frac{12}{4}=3$.

Step2: Apply Chebyshev's theorem

Chebyshev's theorem states that the proportion of data within $k$ standard deviations of the mean is at least $1-\frac{1}{k^{2}}$.
For problem 18:
Substitute $k = 2.4$ into the formula $1-\frac{1}{k^{2}}$.
$1-\frac{1}{2.4^{2}}=1-\frac{1}{5.76}\approx1 - 0.174 = 0.826=82.6\%$.
For problem 19:
Substitute $k = 3$ into the formula $1-\frac{1}{k^{2}}$.
$1-\frac{1}{3^{2}}=1-\frac{1}{9}=\frac{8}{9}\approx0.889 = 88.9\%$.

Answer:

For problem 18: $82.6\%$
For problem 19: $88.9\%$