Question 1

According to the Empirical Rule, for a mound-shaped distribution, approximately what percentage of data falls within two standard deviations of the mean?

Accepted Answer

$$95\%$$

Question 2

A manufacturer claims the average life of a battery is $$60$$ months with a standard deviation of $$10$$ months. Assuming a mound-shaped distribution, what percentage of batteries are expected to last more than $$50$$ months?

Accepted Answer

$$84\%$$

Question 3

For a mound-shaped distribution, which interval contains approximately $$99.7\% of $$the data?

Accepted Answer

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

Question 4

Chebyshev’s Rule states that for any distribution, at least what fraction of the data falls within $$k$$ standard deviations of the mean?

Accepted Answer

$$1 - 1/k^2$

Question 5

A city engineer records the number of vehicles passing through an intersection. The mean is $$375$$ vehicles per day, and the standard deviation is $$25. $$What is the minimum percentage of days with more than $$425$$ vehicles, using Chebyshev’s Rule?

Accepted Answer

$$11.1\%$$

Question 6

If a battery lasts only $$37$$ months, and the manufacturer claims the mean is $$60$$ months with a standard deviation of $$10$$ months, what can you infer about the claim?

Accepted Answer

The claim is questionable, as $$37$$ months is more than two standard deviations below the mean.

Question 7

For a mound-shaped distribution, what is a rough approximation for the standard deviation $$s$$?

Accepted Answer

Sample range divided by $$4$$

Question 8

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

Accepted Answer

It can be applied to any set of data, regardless of distribution shape.

Question 9

Given a mound-shaped distribution with mean $$\mu$$ and standard deviation $$\sigma$$, what is the probability that a randomly selected value is more than $$2\sigma$$ above the mean?

Accepted Answer

$$2.5\%$$

Question 10

If the sample mean is $$100$$ and the sample standard deviation is $$15$$, what is the interval that contains approximately $$68\% of $$the data for a mound-shaped distribution?

Accepted Answer

$$(85, 115)$$

Question 11

A data set has a mean of $$50$$ and a standard deviation of $$5. $$According to Chebyshev’s Rule, at least what percentage of data falls between $$40$$ and $$60$$?

Accepted Answer

$$75\%$$

Question 12

Which of the following is NOT a correct application of the Empirical Rule?

Accepted Answer

Applying it to a highly skewed distribution.

Question 13

If a data set is not mound-shaped, which rule should you use to estimate the minimum percentage of data within $$k$$ standard deviations of the mean?

Accepted Answer

Chebyshev’s Rule

Question 14

A sample has a range of $$40. $$What is the rough approximation for the standard deviation $$s$$ for a mound-shaped distribution?

Accepted Answer

$$10$$

Question 15

For a mound-shaped distribution, what is the approximate probability that a value is more than $$3$$ standard deviations from the mean?

Accepted Answer

$$0.3\%$$

Question 16

If a data set has a mean of $$200$$ and a standard deviation of $$20$$, what is the interval that contains at least $$8/9 of $$the data according to Chebyshev’s Rule?

Accepted Answer

$$(140, 260)$$

Question 17

A data set has a mean of $$80$$ and a standard deviation of $$8. $$According to Chebyshev’s Rule, at least what percentage of data falls within $$3$$ standard deviations of the mean?

Accepted Answer

$$89\%$$

Question 18

A sample of battery lives is mound-shaped with mean $$60$$ months and standard deviation $$10$$ months. What is the approximate percentage of batteries that last less than $$40$$ months?

Accepted Answer

$$2.5\%$$

Question 19

If the mean number of vehicles per day is $$375$$ and the standard deviation is $$25$$, what percentage of days had more than $$425$$ vehicles, assuming a mound-shaped distribution?

Accepted Answer

$$2.5\%$$

Question 20

Which of the following best describes the main difference between the Empirical Rule and Chebyshev’s Rule?

Accepted Answer

Chebyshev’s Rule provides minimum percentages for any distribution, while the Empirical Rule gives approximate percentages for mound-shaped distributions.

Interpreting the Standard Deviation: Empirical Rule and Chebyshev’s Rule

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