Question 1

Chebyshev's Rule applies to any data set, regardless of its distribution. According to Chebyshev's Rule, what is the minimum percentage of measurements that must fall within two standard deviations of the mean for any data set?

Accepted Answer

$$75\%$$

Question 2

The Empirical Rule is used for mound-shaped, symmetric distributions. What percentage of measurements are expected to fall within one standard deviation of the mean according to the Empirical Rule?

Accepted Answer

$$68\%$$

Question 3

A manufacturer claims the average life of its batteries is $$60$$ months with a standard deviation of $$10$$ months, and the distribution is mound-shaped. What percentage of batteries are expected to last less than $$40$$ months?

Accepted Answer

$$16\%$$

Question 4

Suppose you have a sample with mean $$\bar{x} = 50$$ and standard deviation $$s = 5. $$According to Chebyshev's Rule, at least what percentage of measurements must fall within the interval $$[40, 60]$$?

Accepted Answer

$$75\%$$

Question 5

A psychology experiment records the time (in minutes) for $$30$$ rats to run through a maze. If the mean time is $$3.5$$ minutes and the standard deviation is $$1$$ minute, what interval would contain approximately $$99.7\% of $$the times according to the Empirical Rule?

Accepted Answer

$$[0.5, 6.5]$$

Question 6

Which of the following statements is TRUE regarding Chebyshev's Rule for any $$k \u003e 1$$?

Accepted Answer

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

Question 7

If a data set is mound-shaped and symmetric, what is the interval that contains approximately $$95\% of $$the data?

Accepted Answer

$$\bar{x} \pm 2s$$

Question 8

A car manufacturer claims the mean EPA mileage rating for its cars is $$37$$ mpg with a standard deviation of $$2$$ mpg. If the distribution is mound-shaped, what is the expected percentage of cars with mileage between $$35$$ and $$39$$ mpg?

Accepted Answer

$$68\%$$

Question 9

A student claims that in any data set, at least $$89\% of $$the measurements will fall within three standard deviations of the mean. Which rule supports this claim?

Accepted Answer

Chebyshev's Rule

Question 10

If a battery lasts $$37$$ months, and the manufacturer claims the mean is $$60$$ months with a standard deviation of $$10$$ months, what can you infer about the battery's life relative to the distribution?

Accepted Answer

It is more than two standard deviations below the mean, which is unusual.

Question 11

Which of the following intervals would you use to check if a calculated standard deviation is reasonable for a mound-shaped data set?

Accepted Answer

$$\bar{x} \pm s$$, $$\bar{x} \pm 2s$$, $$\bar{x} \pm 3s$$

Question 12

A sample of $$100$$ cars has a mean EPA mileage rating of $$37$$ mpg and a standard deviation of $$2$$ mpg. According to the Empirical Rule, how many cars are expected to have mileage between $$33$$ and $$41$$ mpg?

Accepted Answer

$$95$$

Question 13

Which rule provides a more precise estimate of the percentage of measurements within one, two, and three standard deviations of the mean for mound-shaped distributions?

Accepted Answer

Empirical Rule

Question 14

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

Accepted Answer

Chebyshev's Rule

Question 15

A sample has mean $$\bar{x} = 20$$ and standard deviation $$s = 4. $$What is the interval that contains at least $$75\% of $$the data according to Chebyshev's Rule?

Accepted Answer

$$[12, 28]$$

Question 16

Which of the following is NOT a requirement for using the Empirical Rule?

Accepted Answer

The data must be normally distributed.

Question 17

A data set has a mean of $$50$$ and a standard deviation of $$5. $$According to Chebyshev's Rule, what is the minimum percentage of data within $$[40, 60]$$?

Accepted Answer

$$75\%$$

Question 18

If a sample has a mean $$\bar{x} = 100$$ and standard deviation $$s = 20$$, what interval contains at least $$89\% of $$the data according to Chebyshev's Rule?

Accepted Answer

$$[40, 160]$$

Using the Mean and Standard Deviation to Describe Data: Chebyshev’s Rule and the Empirical Rule

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