Question

given that \\(z_{20} = -2\\) and \\(z_{50} = -1\\), which of the following do you know?

• the variance is 10.
• the standard deviation is 30.
• the mean is 80.
• the median is 40.
• the data point \\(x = 20\\) is 2 standard deviations from the mean.
• the data point \\(x = 50\\) is 1 standard deviation from the mean.
• the data point \\(x = 45\\) has a z-value of 1.5.

Explanation:

Set up the system of equations

Using the Z-Score Formula knowledge point

$$
LATEXBLOCK0
$$

Solve for standard deviation and mean

Using the Z-Score Formula knowledge point

$$
LATEXBLOCK1
$$

Calculate the variance

Using the Z-Score Formula knowledge point

$$
LATEXBLOCK2
$$

Evaluate the given options

Using the Z-Score Formula knowledge point

$$
LATEXBLOCK3
$$

The correct statements are:

• The standard deviation is 30.
• The mean is 80.
• The data point \(x = 20\) is 2 standard deviations from the mean (since \(|z_{20}| = 2\)).
• The data point \(x = 50\) is 1 standard deviation from the mean (since \(|z_{50}| = 1\)).

Answer:

• The variance is 10.
• The standard deviation is 30. (Correct answer)
• The mean is 80. (Correct answer)
• The median is 40.
• The data point \(x = 20\) is 2 standard deviations from the mean. (Correct answer)
• The data point \(x = 50\) is 1 standard deviation from the mean. (Correct answer)
• The data point \(x = 45\) has a z-value of 1.5.