(A)
Obtain the z-score corresponding to \(X = 52\).
The z-score corresponding to \(X = 52\) is obtained below as follows,
From the information given that,
Let X denotes the random variable with the population mean 40 and the standard deviation of 8
Thatis, \(\mu = 40, \sigma = 8\)
The required value is,
\(z=\frac{X-\mu}{\sigma}\)

\(=\frac{52-40}{8}= \frac{12}{8}\)

\(= 1.50\) Thus, the value of the z-score corresponding to \(X = 52 \text{is} 1.50.\) (B) Obtain the X value corresponding to \(z = –0.50\). The X value corresponding to \(z = –0.50\) is obtained below as follows: The required value is, \(z = X - \frac{\mu}{\sigma}\)

\(-0.50 = \frac{X-40}{8}\)

\(-0.50 \times 8.0 = X-40\)

\(-4.00 = X - 40\)

\(X = 40 - 4\)

\(= 36\) Thus, the X value corresponding to \(z = –0.50 \text{is} 36\) It is clear that standard normal distribution represents a nommal curve with mean 0 and standard deviation 1 if the scores in the population are transformed into z-scores with the parameters involved in a nommal distribution are mean \((\mu)\) and standard deviation \((\sigma)\). Determine the value of mean if all of the scores in the population are transformed into z-scores The value of mean if all of the scores in the population are transformed into z-scores is 0. Determine the value of standard deviation if all of the scores in the population are transformed into z-scores. The value of standard deviation if all of the scores in the population are transformed into z-scores is 1.

\(=\frac{52-40}{8}= \frac{12}{8}\)

\(= 1.50\) Thus, the value of the z-score corresponding to \(X = 52 \text{is} 1.50.\) (B) Obtain the X value corresponding to \(z = –0.50\). The X value corresponding to \(z = –0.50\) is obtained below as follows: The required value is, \(z = X - \frac{\mu}{\sigma}\)

\(-0.50 = \frac{X-40}{8}\)

\(-0.50 \times 8.0 = X-40\)

\(-4.00 = X - 40\)

\(X = 40 - 4\)

\(= 36\) Thus, the X value corresponding to \(z = –0.50 \text{is} 36\) It is clear that standard normal distribution represents a nommal curve with mean 0 and standard deviation 1 if the scores in the population are transformed into z-scores with the parameters involved in a nommal distribution are mean \((\mu)\) and standard deviation \((\sigma)\). Determine the value of mean if all of the scores in the population are transformed into z-scores The value of mean if all of the scores in the population are transformed into z-scores is 0. Determine the value of standard deviation if all of the scores in the population are transformed into z-scores. The value of standard deviation if all of the scores in the population are transformed into z-scores is 1.