Questions: For a population with a standard deviation of σ=20, find the z-score for each of the following locations in the distribution. Below the mean by 10 points Below the mean by 30 points Above the mean by 5 points Above the mean by 2 points

Solution Steps

We are tasked with finding the \( z \)-scores for specific points relative to the mean of a normal distribution with a standard deviation of \(\sigma = 20\). The \( z \)-score is calculated using the formula:

\[ z = \frac{{X - \mu}}{\sigma} \]

where \( X \) is the value for which we are calculating the \( z \)-score, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation.

For a point 10 points below the mean, we have:

• \( X = -10 \)
• \(\mu = 0\) (since we are considering deviations from the mean)
• \(\sigma = 20\)

Substituting these values into the formula:

\[ z = \frac{{-10 - 0}}{20} = -0.5 \]

For a point 30 points below the mean, we have:

• \( X = -30 \)
• \(\mu = 0\)
• \(\sigma = 20\)

Substituting these values into the formula:

\[ z = \frac{{-30 - 0}}{20} = -1.5 \]

For a point 5 points above the mean, we have:

• \( X = 5 \)
• \(\mu = 0\)
• \(\sigma = 20\)

Substituting these values into the formula:

\[ z = \frac{{5 - 0}}{20} = 0.25 \]

Final Answer