Question 1

Chebyshev’s Rule states that for any distribution, at least what fraction of the data will fall within 2 standard deviations of the mean?

Accepted Answer

$$\frac{3}{4}$$

Question 2

Which of the following is a correct general statement of Chebyshev’s Rule for any integer $$k \u003e 1$$?

Accepted Answer

At least $$1 - \frac{1}{k^2$} of $$the data will fall within $$k$$ standard deviations of the mean.

Question 3

Suppose a dataset has a mean $$\bar{x} = 375$$ and standard deviation $$s = 25. $$According to Chebyshev’s Rule, at least what percentage of the data falls between $$325$$ and $$425$$?

Accepted Answer

At least $$75\%$$

Question 4

If nothing is known about the shape of a distribution, which rule can be used to estimate the minimum proportion of data within $$k$$ standard deviations of the mean?

Accepted Answer

Chebyshev’s Rule

Question 5

For a mound-shaped (normal) distribution, what percentage of data falls within 2 standard deviations of the mean?

Accepted Answer

$$95\%$$

Question 6

A city engineer records the number of vehicles passing through an intersection each day. The mean is $$375$$ and the standard deviation is $$25. $$What is the minimum percentage of days with vehicle counts between $$325$$ and $$425$$ according to Chebyshev’s Rule?

Accepted Answer

$$75\%$$

Question 7

Chebyshev’s Rule can be applied to which type of data distributions?

Accepted Answer

Any distribution, regardless of shape

Question 8

If $$k = 3$$, what is the minimum proportion of data within 3 standard deviations of the mean according to Chebyshev’s Rule?

Accepted Answer

$$\frac{8}{9}$$

Question 9

A dataset has mean $$\bar{x}$$ and standard deviation $$s. $$According to Chebyshev’s Rule, what is the interval that contains at least $$1 - \frac{1}{k^2$} of $$the data?

Accepted Answer

$$(\bar{x} - ks, \bar{x} + ks)$$

Question 10

If a distribution is mound-shaped, which rule provides a more accurate estimate of the percentage of data within 2 standard deviations of the mean?

Accepted Answer

Empirical Rule

Question 11

Given a mean of $$375$$ and standard deviation of $$25$$, what is the $$z-$$score for a day with $$425$$ vehicles?

Accepted Answer

$$z = 2$$

Question 12

If you know nothing about the shape of a dataset, what is the maximum percentage of data that could fall outside 2 standard deviations from the mean according to Chebyshev’s Rule?

Accepted Answer

$$25\%$$

Question 13

Which of the following is NOT a correct application of Chebyshev’s Rule?

Accepted Answer

Estimating the exact proportion of data within $$k$$ standard deviations for a normal distribution

Question 14

A dataset has a mean $$\bar{x}$$ and standard deviation $$s. $$For $$k = 4$$, what is the minimum percentage of data within 4 standard deviations of the mean according to Chebyshev’s Rule?

Accepted Answer

$$93.75\%$$

Question 15

Which of the following statements is true about Chebyshev’s Rule?

Accepted Answer

It provides a minimum percentage of data within $$k$$ standard deviations for any distribution.

Question 16

If a dataset is mound-shaped, what percentage of data falls within 1 standard deviation of the mean according to the Empirical Rule?

Accepted Answer

$$68\%$$

Question 17

A city engineer wants to estimate the percentage of days with more than 425 vehicles, given a mean of 375 and standard deviation of 25. If the distribution is mound-shaped, what rule should be used?

Accepted Answer

Empirical Rule

Question 18

If $$k = 2$$, what is the minimum proportion of data within 2 standard deviations of the mean according to Chebyshev’s Rule?

Accepted Answer

$$\frac{3}{4}$$

Question 19

Which of the following intervals contains at least $$89\% of $$the data according to Chebyshev’s Rule?

Accepted Answer

$$(\bar{x} - 3s, \bar{x} + 3s)$$

Question 20

Which of the following is a limitation of Chebyshev’s Rule?

Accepted Answer

It only provides a minimum, not an exact, percentage of data within $$k$$ standard deviations.

Chebyshev’s Rule and Its Application in Statistics

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