Chebyshev's theorem states that for any distribution (regardless of its shape), at least $\frac{1}{k^2}$ of the data values will fall within $k$ standard deviations from the mean, where $k > 1$.

In this problem, we know the following:

• The mean salary ($\mu$) is 50,000.
• The standard deviation ($\sigma$) is 10,000.

We are asked to find the range of salaries within which at least 75% of the workers will fall.

Step 1: Determine $k$ for 75% using Chebyshev's theorem

According to Chebyshev's theorem, the proportion of data within $k$ standard deviations of the mean is at least $1 - \frac{1}{k^2}$.

Setting this equal to 0.75:

$1 - \frac{1}{k^2} = 0.75$

$\frac{1}{k^2} = 0.25$

$k^2 = 4$

$k = 2$

Step 2: Calculate the range

Using $k = 2$, we find the range of salaries by adding and subtracting $2 \times \sigma$ from the mean:

• The mean ($\mu$) is 50,000.
• The standard deviation ($\sigma$) is 10,000.

$\text{Lower limit} = \mu - 2\sigma = 50,000 - 2 \times 10,000 = 50,000 - 20,000 = 30,000$

$\text{Upper limit} = \mu + 2\sigma = 50,000 + 2 \times 10,000 = 50,000 + 20,000 = 70,000$

Conclusion

Using Chebyshev's theorem, we can conclude that at least 75% of the workers will have salaries that fall within the range of 30,000 to 70,000.

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Would you like more details or have any questions?

Here are some related questions to consider:

1. What is Chebyshev's theorem and how is it used in statistics?
2. How would the range change if we wanted to find the salary range for 89% of the workers?
3. How does Chebyshev's theorem differ from the Empirical Rule?
4. Can Chebyshev's theorem be applied to any type of distribution, and why?
5. What is the importance of knowing the standard deviation in a set of data?

Tip: Chebyshev’s theorem is very useful when dealing with non-normal distributions because it makes no assumptions about the shape of the data distribution.