Q: Suppose the mean height of a group of men is $$70$$ inches with a standard deviation of $$2$$ inches. What is the height corresponding to a z-score of $$-1.5$$?

$$67$$ inches$$\ (70 + (-1.5) \times 2)$$

Question 1

According to the Empirical Rule, for a unimodal and symmetric distribution, approximately what percentage of data falls within two standard deviations of the mean?

Accepted Answer

$$95\%$$

Question 2

A sample of cities has a mean smog level of $$7.9$$ micrograms per cubic meter and a standard deviation of $$2.2. $$According to the Empirical Rule, what is the range that contains nearly all cities' smog levels?

Accepted Answer

$$1.3 to$$ $$14.5$$

Question 3

Which of the following best describes a z-score?

Accepted Answer

The number of standard deviations a value is from the mean.

Question 4

Given a mean of $$70$$ and a standard deviation of $$10$$, what is the z-score for a value of $$90$$?

Accepted Answer

$$z = \frac{90 - 70}{10} = 2$$

Question 5

Maria scored $$80 on $$Exam 1 (mean $$70$$, SD $$10) $$and $$85 on $$Exam 2 (mean $$80$$, SD $$5). On $$which exam did she perform better relative to her class?

Accepted Answer

She performed equally well on both exams.

Question 6

If a data value has a z-score of $$-2.5$$, what does this indicate?

Accepted Answer

The value is $$2.5$$ standard deviations below the mean.

Question 7

A distribution of daily high temperatures in San Francisco is unimodal and symmetric. If $$95\% of $$the temperatures are between $$x$$ and $$y$$, what percentage of days are colder than $$x$$?

Accepted Answer

$$2.5\%$$

Question 8

Which scenario is most appropriate for using the Empirical Rule?

Accepted Answer

A distribution that is unimodal and symmetric.

Question 9

If a student’s z-score on a test is $$0$$, what does this mean?

Accepted Answer

The student scored exactly at the mean.

Question 10

Suppose the mean height of a group of men is $$70$$ inches with a standard deviation of $$2$$ inches. What is the height corresponding to a z-score of $$-1.5$$?

Accepted Answer

$$67$$ inches$$\ (70 + (-1.5) 	imes 2)$$

Question 11

A city has a smog level of $$12.3$$ micrograms per cubic meter. If the mean is $$7.9$$ and the standard deviation is $$2.2$$, what is the z-score for this city?

Accepted Answer

$$z = \frac{12.3 - 7.9}{2.2} = 2$$

Question 12

Why are z-scores useful in comparing values from different distributions?

Accepted Answer

They convert values to a common scale based on standard deviations.

Question 13

If a value is more than $$3$$ standard deviations from the mean in a symmetric, unimodal distribution, what can you conclude?

Accepted Answer

It is likely to be an outlier or unusual.

Question 14

A data set is unimodal and symmetric with mean $$50$$ and standard deviation $$8. $$What is the interval that contains about $$68\% of $$the data?

Accepted Answer

$$42 to$$ $$58$$

Question 15

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

Accepted Answer

Calculating the exact probability of an event in a skewed distribution.

Numerical Summaries of Center and Variation: The Empirical Rule and z-Scores

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