Question 1

Which of the following best describes the normal curve?

Accepted Answer

A bell-shaped curve that is unimodal, symmetric, and defined mathematically.

Question 2

As the sample size increases, what happens to the shape of the sample distribution?

Accepted Answer

It becomes more like the normal curve.

Question 3

Which formula is used to calculate a z score for a raw score $$X in a $$population with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$z = \frac{X - \mu}{\sigma}$$

Question 4

If a dog is much larger than the average dog, what can you say about its z score for size?

Accepted Answer

It will have a positive z score.

Question 5

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

Accepted Answer

It is only used for nominal (categorical) variables.

Question 6

How do you transform a z score back into a raw score $$X$$?

Accepted Answer

$$X = z \sigma + \mu$$

Question 7

Why is standardization useful when comparing scores from different distributions?

Accepted Answer

It allows scores to be compared directly by converting them to a common scale.

Question 8

What does the Central Limit Theorem state about the distribution of sample means?

Accepted Answer

Sample means are normally distributed even if the population is not normal.

Question 9

Which formula gives the standard error of the mean for a population with standard deviation $$\sigma$$ and sample size $$N$$?

Accepted Answer

$$\sigma_M = \frac{\sigma}{\sqrt{N}}$$

Question 10

A researcher wants to compare test scores from two different exams with different means and standard deviations. What should they do?

Accepted Answer

Convert both sets of scores to z scores before comparing.

Question 11

If a score has a z value of $$-2$$, what does this indicate about its position in the distribution?

Accepted Answer

It is two standard deviations below the mean.

Question 12

Which of the following statements about the area under the normal curve is correct?

Accepted Answer

The area under the curve can be used to calculate percentiles for any score.

Question 13

What is the main difference between a distribution of scores and a distribution of means?

Accepted Answer

A distribution of means is less variable than a distribution of scores.

Question 14

Which formula should be used to calculate the z score for a sample mean $$M$$ when the population mean is $$\mu_M$$ and the standard error is $$\sigma_M$$?

Accepted Answer

$$z = \frac{M - \mu_M}{\sigma_M}$$

Question 15

A student receives a z score of 1.5 on a standardized test. What does this mean?

Accepted Answer

The student's score is 1.5 standard deviations above the mean.

Question 16

Which of the following is a real-world application of z scores in research?

Accepted Answer

Comparing the prevalence of emotions across different years using standardized scores.

Question 17

If a distribution is severely skewed, what effect does increasing the sample size have on the distribution of means?

Accepted Answer

It makes the distribution of means more normal.

Question 18

Why can't nominal (categorical) variables be normally distributed?

Accepted Answer

Because they do not have a meaningful order or scale for calculating means and standard deviations.

Question 19

What percentage of scores fall within one standard deviation of the mean in a normal distribution?

Accepted Answer

Approximately 68%

Question 20

A teacher suspects cheating because a student's test score is an extreme outlier. Which statistical tool could help identify this?

Accepted Answer

Calculating the student's z score to see how far it is from the mean.

The Normal Curve, Standardization, and z Scores

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