Question

a variable ( x ) is normally distributed with mean 16 and standard deviation 3. round your answers to the nearest hundredth as needed.

a) determine the ( z )-score for ( x = 23 ).

( z = square )

b) determine the ( z )-score for ( x = 11 ).

( z = square )

c) what value of ( x ) has a ( z )-score of 3?

( x = square )

d) what value of ( x ) has a ( z )-score of ( -1 )?

( x = square )

e) what value of ( x ) has a ( z )-score of 0?

( x = square )

question help: (\boxed{\text{video}}) (\boxed{\text{message instructor}})

Explanation:

Step1: Recall the z - score formula

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation. Given \(\mu = 16\) and \(\sigma=3\).

Step2: Calculate the z - score for \(x = 23\)

Substitute \(x = 23\), \(\mu = 16\), and \(\sigma = 3\) into the formula: \(z=\frac{23 - 16}{3}=\frac{7}{3}\approx2.33\).

Step3: Calculate the z - score for \(x = 11\)

Substitute \(x = 11\), \(\mu = 16\), and \(\sigma = 3\) into the formula: \(z=\frac{11 - 16}{3}=\frac{-5}{3}\approx - 1.67\).

Step4: Rearrange the z - score formula to solve for \(x\)

From \(z=\frac{x-\mu}{\sigma}\), we can get \(x=\mu+z\sigma\).

Step5: Find \(x\) when \(z = 3\)

Substitute \(z = 3\), \(\mu = 16\), and \(\sigma = 3\) into \(x=\mu+z\sigma\): \(x=16+3\times3=16 + 9=25\).

Step6: Find \(x\) when \(z=-1\)

Substitute \(z=-1\), \(\mu = 16\), and \(\sigma = 3\) into \(x=\mu+z\sigma\): \(x=16+( - 1)\times3=16-3 = 13\).

Step7: Find \(x\) when \(z = 0\)

Substitute \(z = 0\), \(\mu = 16\), and \(\sigma = 3\) into \(x=\mu+z\sigma\): \(x=16+0\times3=16\).

Answer:

a) \(z\approx2.33\)
b) \(z\approx - 1.67\)
c) \(x = 25\)
d) \(x = 13\)
e) \(x = 16\)