AP Stats 2.1 Z-scores and Linear Transformation

What is the z-score of a data point that is 10 units above the mean, given that the standard deviation is 5?

• 1

• 2

• 0.5

• 5

Answer explanation

The z-score is calculated as (X - mean) / standard deviation. Here, X is 10, mean is 0, and standard deviation is 5. Thus, z = (10 - 0) / 5 = 2. Therefore, the correct answer is 2.

According to the empirical rule, what percentage of data falls within one standard deviation of the mean in a normal distribution?

• 50%

• 68%

• 95%

• 99.7%

Answer explanation

According to the empirical rule, in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. This is a key characteristic of normal distributions.

If a z-score is 0, what does this indicate about the data point?

• It is below the mean.

• It is above the mean.

• It is equal to the mean.

• It is an outlier.

Answer explanation

A z-score of 0 indicates that the data point is exactly at the mean of the distribution. Therefore, the correct choice is that it is equal to the mean.

A dataset has a mean of 20 and a standard deviation of 4. What is the z-score of a data point that is 28?

• 1

• 2

• 3

• 4

Answer explanation

To find the z-score, use the formula: z = (X - mean) / standard deviation. Here, z = (28 - 20) / 4 = 8 / 4 = 2. Thus, the z-score of 28 is 2, which is the correct answer.

If a data point has a z-score of -1.5, what does this indicate?

• The data point is 1.5 standard deviations above the mean.

• The data point is 1.5 standard deviations below the mean.

• The data point is at the mean.

• The data point is an outlier.

Answer explanation

A z-score of -1.5 indicates that the data point is 1.5 standard deviations below the mean. This means it is lower than the average value in the dataset.

According to the empirical rule, approximately what percentage of data falls within three standard deviations of the mean?

• 68%

• 95%

• 99.7%

• 100%

Answer explanation

According to the empirical rule, approximately 99.7% of data falls within three standard deviations of the mean in a normal distribution. This makes 99.7% the correct choice.

• The mean is multiplied by 3.

• The mean is increased by 2.

• The mean is multiplied by 3 and increased by 2.

• The mean remains unchanged.

Answer explanation

A linear transformation of the form y = 3x + 2 scales the mean of the dataset by 3 and then adds 2. Therefore, the correct choice is that the mean is multiplied by 3 and increased by 2.

A dataset has a mean of 50 and a standard deviation of 10. What is the z-score of a data point that is 40?

• -1

• 1

• -0.5

• 0.5

Answer explanation

To find the z-score, use the formula: z = (X - mean) / standard deviation. Here, X = 40, mean = 50, and standard deviation = 10. Thus, z = (40 - 50) / 10 = -10 / 10 = -1. Therefore, the correct answer is -1.

• It is doubled.

• It is increased by 3.

• It remains unchanged.

• It is halved.

Answer explanation

When transforming a dataset with the equation y = 2x + 3, the standard deviation is affected by the coefficient of x. Since the coefficient is 2, the standard deviation is doubled, making 'It is doubled' the correct choice.

According to the empirical rule, what percentage of data falls within two standard deviations of the mean?

• 68%

• 95%

• 99.7%

• 100%

Answer explanation

According to the empirical rule, approximately 95% of data falls within two standard deviations of the mean in a normal distribution. Therefore, the correct answer is 95%.