Question

q#1: given the population mean \\(\mu = 18\\), population standard deviation \\(\sigma = 4\\) and a data set below find the z score only for 15 from the list.
data set: 12, 13, 15, 9, 4, 10, 30, 20 and 60.
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q#2: find the z-score, given the population mean \\(\mu = 78\\), population standard deviation \\(\sigma = 10\\) an x value = 70
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q#3: convert this non-standard curve to a standard normal curve. make sure you leveled the z-score clearly!
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q#4: find the data value x, given \\(\mu = 95\\), \\(\sigma = 8\\) & \\(z = -3.25\\)
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q#5: find the standard deviation given \\(\mu = 75\\), \\(x = 85\\) & \\(z = 2.89\\).

Explanation:

Identify given values and formula

We use the z-score formula:

$$z = \frac{X - \mu}{\sigma}$$

For Q#1: \(\mu = 18\), \(\sigma = 4\), and \(X = 15\).

Calculate z-score for Q#1

Substitute the values into the formula:

$$z = \frac{15 - 18}{4} = \frac{-3}{4} = -0.75$$

Calculate z-score for Q#2

For Q#2: \(\mu = 78\), \(\sigma = 10\), and \(X = 70\).

$$z = \frac{70 - 78}{10} = \frac{-8}{10} = -0.8$$

Convert curve for Q#3

The mean \(\mu = 60\) is at the center.
The standard deviation \(\sigma\) is the distance between tick marks:

$$\sigma = 72 - 60 = 12$$

To convert to a standard normal curve, the mean \(\mu\) becomes \(z = 0\).
Each tick mark represents one standard deviation:

• \(60

ightarrow 0\)

• \(72

ightarrow 1\)

• \(84

ightarrow 2\)

• \(96

ightarrow 3\)

• \(48

ightarrow -1\)

• \(36

ightarrow -2\)

• \(24

ightarrow -3\)

Solve for X in Q#4

For Q#4: \(\mu = 95\), \(\sigma = 8\), and \(Z = -3.25\).
Rearrange the formula to solve for \(X\):

$$X = \mu + Z \cdot \sigma$$

$$X = 95 + (-3.25) \cdot 8 = 95 - 26 = 69$$

Solve for standard deviation in Q#5

For Q#5: \(\mu = 75\), \(x = 85\), and \(Z = 2.89\).
Rearrange the formula to solve for \(\sigma\):

$$\sigma = \frac{x - \mu}{Z}$$

$$\sigma = \frac{85 - 75}{2.89} = \frac{10}{2.89} \approx 3.46$$

Answer:

Question 1

The z-score for 15 is:

$$z = -0.75$$

Question 2

The z-score is:

$$z = -0.8$$

Question 3

The standard normal curve has a mean of \(0\) and a standard deviation of \(1\). The corresponding z-scores for the labeled points on the horizontal axis are:

• \(24

ightarrow -3\)

• \(36

ightarrow -2\)

• \(48

ightarrow -1\)

• \(60

ightarrow 0\)

• \(72

ightarrow 1\)

• \(84

ightarrow 2\)

• \(96

ightarrow 3\)

Question 4

The data value \(X\) is:

$$X = 69$$

Question 5

The standard deviation \(\sigma\) is:

$$\sigma \approx 3.46$$