Answer :
To find the [tex]\( z \)[/tex]-score of the given value [tex]\( x = 7.5 \)[/tex] for a normally distributed variable [tex]\( X \sim N(5.5, 2) \)[/tex], follow these steps:
1. Identify the Mean ([tex]\(\mu\)[/tex]) and Standard Deviation ([tex]\(\sigma\)[/tex]):
- The mean ([tex]\(\mu\)[/tex]) of the normal distribution is 5.5.
- The standard deviation ([tex]\(\sigma\)[/tex]) is 2.
2. Use the [tex]\( z \)[/tex]-score Formula:
The formula to calculate the [tex]\( z \)[/tex]-score is:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
where:
- [tex]\( x \)[/tex] is the value you are analyzing (in this case, 7.5).
- [tex]\(\mu\)[/tex] is the mean of the distribution.
- [tex]\(\sigma\)[/tex] is the standard deviation.
3. Plug in the Values:
Substitute the known values into the formula:
[tex]\[
z = \frac{7.5 - 5.5}{2}
\][/tex]
4. Calculate:
Simplify the expression:
[tex]\[
z = \frac{2}{2} = 1
\][/tex]
5. Interpret the [tex]\( z \)[/tex]-score:
A [tex]\( z \)[/tex]-score of 1 means that the value [tex]\( x = 7.5 \)[/tex] is one standard deviation ([tex]\(1 \sigma\)[/tex]) above or to the right of the mean ([tex]\(\mu = 5.5\)[/tex]).
Therefore, the correct interpretation is:
"This means that [tex]\( x = 7.5 \)[/tex] is one standard deviation ([tex]\(1 \sigma\)[/tex]) above or to the right of the mean, [tex]\( \mu = 5.5 \)[/tex]."
1. Identify the Mean ([tex]\(\mu\)[/tex]) and Standard Deviation ([tex]\(\sigma\)[/tex]):
- The mean ([tex]\(\mu\)[/tex]) of the normal distribution is 5.5.
- The standard deviation ([tex]\(\sigma\)[/tex]) is 2.
2. Use the [tex]\( z \)[/tex]-score Formula:
The formula to calculate the [tex]\( z \)[/tex]-score is:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
where:
- [tex]\( x \)[/tex] is the value you are analyzing (in this case, 7.5).
- [tex]\(\mu\)[/tex] is the mean of the distribution.
- [tex]\(\sigma\)[/tex] is the standard deviation.
3. Plug in the Values:
Substitute the known values into the formula:
[tex]\[
z = \frac{7.5 - 5.5}{2}
\][/tex]
4. Calculate:
Simplify the expression:
[tex]\[
z = \frac{2}{2} = 1
\][/tex]
5. Interpret the [tex]\( z \)[/tex]-score:
A [tex]\( z \)[/tex]-score of 1 means that the value [tex]\( x = 7.5 \)[/tex] is one standard deviation ([tex]\(1 \sigma\)[/tex]) above or to the right of the mean ([tex]\(\mu = 5.5\)[/tex]).
Therefore, the correct interpretation is:
"This means that [tex]\( x = 7.5 \)[/tex] is one standard deviation ([tex]\(1 \sigma\)[/tex]) above or to the right of the mean, [tex]\( \mu = 5.5 \)[/tex]."
