Answer :
To calculate the z-score for the given value of [tex]x = 29.5[/tex], we'll use the formula for the z-score, which is:
[tex]z = \frac{x - \mu}{\sigma}[/tex]
Where:
- [tex]x[/tex] is the value for which the z-score is being calculated,
- [tex]\mu[/tex] is the mean of the sample,
- [tex]\sigma[/tex] is the standard deviation of the sample.
Given values:
- Sample mean [tex]\mu = 35.8[/tex]
- Sample standard deviation [tex]\sigma = 6.8[/tex]
- [tex]x = 29.5[/tex]
Now, substitute these values into the z-score formula:
[tex]z = \frac{29.5 - 35.8}{6.8}[/tex]
Calculate the difference in the numerator:
[tex]29.5 - 35.8 = -6.3[/tex]
Substitute this back into the equation:
[tex]z = \frac{-6.3}{6.8}[/tex]
Now, divide the numerator by the standard deviation:
[tex]z \approx -0.9265[/tex]
Round this value to two decimal places:
[tex]z \approx -0.93[/tex]
Therefore, the z-score for [tex]x = 29.5[/tex] is approximately [tex]-0.93[/tex].
This z-score indicates that the value of 29.5 is approximately 0.93 standard deviations below the mean of 35.8 in this data set. A negative z-score shows that the value is below the mean.
