Question 1

Which of the following best describes the purpose of standardizing data using z-scores?

Accepted Answer

To compare values from different data sets by expressing them in terms of standard deviations from the mean.

Question 2

Given a data set with mean $$\overline{y} = 50$$ and standard deviation $$s = 10$$, what is the z-score for a value $$y = 70$$?

Accepted Answer

$$z = 2$$

Question 3

Which of the following statements about the Normal Model is TRUE?

Accepted Answer

The Normal Model is symmetric and bell-shaped, and is used to approximate the distribution of many quantitative variables.

Question 4

A researcher multiplies every value in a data set by 3. What effect does this have on the standard deviation?

Accepted Answer

The standard deviation is multiplied by 3.

Question 5

According to the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

95%

Question 6

If a data set is standardized, what will be the mean and standard deviation of the resulting z-scores?

Accepted Answer

Mean = 0, Standard deviation = 1

Question 7

A student scores 1.5 standard deviations above the mean on a standardized test. What is their z-score?

Accepted Answer

$$z = 1.5$$

Question 8

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is skewed to the right.

Question 9

Suppose the heights of adult men are normally distributed with mean $$\mu = 70$$ inches and standard deviation $$\sigma = 3$$ inches. What is the probability that a randomly selected man is taller than 76 inches?

Accepted Answer

About 2.5%

Question 10

Which of the following best explains why standardizing data is useful when comparing test scores from different exams (e.g., SAT and ACT)?

Accepted Answer

It allows scores to be compared on a common scale, regardless of the original units or distributions.

Question 11

A histogram of a data set is symmetric and bell-shaped. Which model is most appropriate for describing this data?

Accepted Answer

Normal Model

Question 12

If a data set is shifted by subtracting a constant from every value, what happens to the standard deviation?

Accepted Answer

It remains unchanged.

Question 13

Which of the following is the correct formula for calculating a z-score?

Accepted Answer

$$z = \frac{y - \overline{y}}{s}$$

Question 14

A normal probability plot is used to:

Accepted Answer

Check whether a data set is approximately normal.

Question 15

If a data set is unimodal and symmetric but has heavy tails, which of the following is true?

Accepted Answer

The normal model may not be a perfect fit, but it could still be used as an approximation.

Question 16

Which of the following best describes the effect of scaling data by a constant factor?

Accepted Answer

All measures of spread (such as standard deviation) are multiplied by the same constant.

Question 17

A z-score of $$-2$$ indicates that the data value is:

Accepted Answer

Two standard deviations below the mean.

Question 18

Which of the following is NOT a benefit of standardizing data?

Accepted Answer

It changes the shape of the data distribution to normal.

Question 19

If a normal probability plot shows a strong curve rather than a straight line, what does this indicate about the data?

Accepted Answer

The data are not normally distributed.

Question 20

A data value has a z-score of $$0. $$What does this mean?

Accepted Answer

The value is equal to the mean.

Standard Deviation as a Ruler and the Normal Model: Study Notes

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