Sure! To find the z-score for a given value in a dataset, you can use the formula for the z-score, which is given by:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
where:
- \( x \) is the value for which you are calculating the z-score,
- \( \mu \) is the mean of the dataset,
- \( \sigma \) is the standard deviation of the dataset.
Given:
- Mean (\(\mu\)) = 3977
- Standard deviation (\(\sigma\)) = 106
(c) To find the z-score for \( x = 4300 \):
[tex]\[ z_c = \frac{4300 - 3977}{106} \][/tex]
Evaluating this:
- Subtract the mean from the value: \( 4300 - 3977 = 323 \)
- Divide by the standard deviation: \( \frac{323}{106} \approx 3.047 \)
Therefore, the z-score for \( x = 4300 \) is approximately \( 3.047 \).
(d) To find the z-score for \( x = 4000 \):
[tex]\[ z_d = \frac{4000 - 3977}{106} \][/tex]
Evaluating this:
- Subtract the mean from the value: \( 4000 - 3977 = 23 \)
- Divide by the standard deviation: \( \frac{23}{106} \approx 0.217 \)
Therefore, the z-score for \( x = 4000 \) is approximately \( 0.217 \).
So the z-scores are:
(c) \( z \) for \( x = 4300 \) is \( \boxed{3.047} \)
(d) [tex]\( z \)[/tex] for [tex]\( x = 4000 \)[/tex] is [tex]\( \boxed{0.217} \)[/tex]
