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Answer :
To calculate the standard score (also known as the z-score) for the given value [tex]\( X = 98.1 \)[/tex], with a mean ([tex]\(\mu\)[/tex]) of 107.8 and a standard deviation ([tex]\(\sigma\)[/tex]) of 106, you can follow these steps:
1. Understand the Formula for the Z-score:
The z-score is calculated using the formula:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
Here:
- [tex]\( X \)[/tex] is the value for which you want to find the z-score.
- [tex]\( \mu \)[/tex] is the mean of the dataset.
- [tex]\( \sigma \)[/tex] is the standard deviation of the dataset.
2. Plug in the Given Values:
Substitute the given numbers into the formula:
[tex]\[
z = \frac{98.1 - 107.8}{106}
\][/tex]
3. Calculate the Difference:
First, calculate the difference between [tex]\( X \)[/tex] and [tex]\( \mu \)[/tex]:
[tex]\[
98.1 - 107.8 = -9.7
\][/tex]
4. Divide by the Standard Deviation:
Next, divide the result by the standard deviation:
[tex]\[
z = \frac{-9.7}{106} \approx -0.09150943396226417
\][/tex]
5. Round the Result:
Finally, round the z-score to two decimal places:
[tex]\[
z \approx -0.09
\][/tex]
So, the standard score (z-score) for [tex]\( X = 98.1 \)[/tex] is approximately [tex]\( -0.09 \)[/tex].
1. Understand the Formula for the Z-score:
The z-score is calculated using the formula:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
Here:
- [tex]\( X \)[/tex] is the value for which you want to find the z-score.
- [tex]\( \mu \)[/tex] is the mean of the dataset.
- [tex]\( \sigma \)[/tex] is the standard deviation of the dataset.
2. Plug in the Given Values:
Substitute the given numbers into the formula:
[tex]\[
z = \frac{98.1 - 107.8}{106}
\][/tex]
3. Calculate the Difference:
First, calculate the difference between [tex]\( X \)[/tex] and [tex]\( \mu \)[/tex]:
[tex]\[
98.1 - 107.8 = -9.7
\][/tex]
4. Divide by the Standard Deviation:
Next, divide the result by the standard deviation:
[tex]\[
z = \frac{-9.7}{106} \approx -0.09150943396226417
\][/tex]
5. Round the Result:
Finally, round the z-score to two decimal places:
[tex]\[
z \approx -0.09
\][/tex]
So, the standard score (z-score) for [tex]\( X = 98.1 \)[/tex] is approximately [tex]\( -0.09 \)[/tex].
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