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.
e. for part a, detena population. choose the correct answer below.
the z - score locates2 standard deviations below the mean.
for part b, determipopulation. choose the correct answer below.
the z - score locates0.5 standard deviations below the mean.
for part c, determipopulation. choose the correct answer below.
the z - score locates2 standard deviations above the mean.
for part d, determipopulation. choose the correct answer below.
the z - score locatesat the mean.
0.5 standard deviations above the mean.
1.2 standard deviations below the mean.
f. for part a, state1.2 standard deviations above the mean.
the value of x lies

Explanation:

Step1: Recall the z - score formula

The z - score formula for a sample is \(z=\frac{x - \bar{x}}{s}\) and for a population is \(z=\frac{x-\mu}{\sigma}\)

Step2: Calculate z - scores for each part

• Part a:

Using the sample z - score formula \(z=\frac{x - \bar{x}}{s}\), with \(x = 28\), \(\bar{x}=20\), \(s = 4\)
\(z=\frac{28 - 20}{4}=\frac{8}{4}=2\)

• Part b:

Using the population z - score formula \(z=\frac{x-\mu}{\sigma}\), with \(x = 88\), \(\mu=87\), \(\sigma = 2\)
\(z=\frac{88 - 87}{2}=\frac{1}{2}=0.5\)

• Part c:

Using the population z - score formula \(z=\frac{x-\mu}{\sigma}\), with \(x = 45\), \(\mu=45\), \(\sigma = 5\)
\(z=\frac{45 - 45}{5}=0\)

• Part d:

Using the sample z - score formula \(z=\frac{x - \bar{x}}{s}\), with \(x = 35\), \(\bar{x}=50\), \(s = 6\)
\(z=\frac{35 - 50}{6}=\frac{- 15}{6}=-2.5\)

Step3: Determine sample or population for each part

• Part a:

Since we use \(\bar{x}\) (sample mean) and \(s\) (sample standard deviation), it is a sample.

• Part b:

Since we use \(\mu\) (population mean) and \(\sigma\) (population standard deviation), it is a population.

• Part c:

Since we use \(\mu\) (population mean) and \(\sigma\) (population standard deviation), it is a population.

• Part d:

Since we use \(\bar{x}\) (sample mean) and \(s\) (sample standard deviation), it is a sample.

Step4: Determine position relative to the mean

• Part a:

\(z = 2>0\), so \(x\) lies \(2\) standard deviations above the mean.

• Part b:

\(z=0.5>0\), so \(x\) lies \(0.5\) standard deviations above the mean.

• Part c:

\(z = 0\), so \(x\) lies at the mean.

• Part d:

\(z=-2.5<0\), so \(x\) lies \(2.5\) standard deviations below the mean.

Answer:

• a: \(z = 2\), sample, \(2\) standard deviations above the mean.
• b: \(z = 0.5\), population, \(0.5\) standard deviations above the mean.
• c: \(z = 0\), population, at the mean.
• d: \(z=-2.5\), sample, \(2.5\) standard deviations below the mean.