Question
The average playing time of albums in a certain collection is 27 minutes, with a standard deviation of 4 minutes. Calculate the z-score, to the nearest thousandth, for an album that has a playing time of 29.5 minutes.
Answers
GPT-4o mini
To calculate the z-score for the album with a playing time of 29.5 minutes, we will use the formula for the z-score:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) is the value we are interested in (29.5 minutes),
- \(\mu\) is the mean (average playing time, 27 minutes),
- \(\sigma\) is the standard deviation (4 minutes).
Now, substituting the values into the formula:
\[
z = \frac{(29.5 - 27)}{4}
\]
Calculating the numerator:
\[
29.5 - 27 = 2.5
\]
Now dividing by the standard deviation:
\[
z = \frac{2.5}{4} = 0.625
\]
To the nearest thousandth, the z-score is:
\[
\boxed{0.625}
\]
\[
z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) is the value we are interested in (29.5 minutes),
- \(\mu\) is the mean (average playing time, 27 minutes),
- \(\sigma\) is the standard deviation (4 minutes).
Now, substituting the values into the formula:
\[
z = \frac{(29.5 - 27)}{4}
\]
Calculating the numerator:
\[
29.5 - 27 = 2.5
\]
Now dividing by the standard deviation:
\[
z = \frac{2.5}{4} = 0.625
\]
To the nearest thousandth, the z-score is:
\[
\boxed{0.625}
\]