Question

compute the z - score corresponding to each of the values of x below.
a. x = 28, s = 4, \\( \overline { x } = 20 \\) b. x = 88, \\( \mu = 87 \\), \\( \sigma = 2 \\) c. \\( \mu = 45 \\), \\( \sigma = 5 \\), x = 45
d. s = 6, x = 35, \\( \overline { x } = 50 \\) e. in parts a - d, state whether the z - score locates x within a sample or a population. f. in parts a - d, state whether each value of x lies above or below the mean and by how many standard deviations.
c. z = 0 (simplify your answer.)
d. z = - 2.5 (simplify your answer.)
e. for part a, determine whether the z - score locates x within a sample or a population. choose the correct answer below.
the z - score locates x within a sample.
for part b, determine whether the z - score locates x within a sample or a population. choose the correct answer below.
the z - score locates x within a
for part c, determine whether thin a sample or a population. choose the correct answer below.
the z - score locates x within a population.
for part d, determine whether within a sample or a population. choose the correct answer below.
the z - score locates x within a

Explanation:

Step1: Recall the formula for z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\) for a population and \(z = \frac{x-\bar{x}}{s}\) for a sample, where \(x\) is the data point, \(\mu\) (or \(\bar{x}\)) is the mean, and \(\sigma\) (or \(s\)) is the standard deviation.

Step2: Determine for part b

In part b, we have \(\mu = 87\) (population mean) and \(\sigma=2\) (population standard deviation). So, using the formula \(z=\frac{x-\mu}{\sigma}\), the z - score formula for a population is applicable.

Step3: Determine for part d

In part d, we have \(\bar{x}=50\) (sample mean) and \(s = 6\) (sample standard deviation). So, using the formula \(z=\frac{x-\bar{x}}{s}\), the z - score formula for a sample is applicable.

Answer:

For part b: The z - score locates \(x\) within a population.
For part d: The z - score locates \(x\) within a sample.