Question

the mean of a set of credit scores is \\( \mu = 690 \\) and \\( \sigma = 14 \\). which statement must be true about \\( z_{694} \\)?
\\( z_{694} \\) is within 1 standard deviation of the mean.
\\( z_{694} \\) is between 1 and 2 standard deviations of the mean.
\\( z_{694} \\) is between 2 and 3 standard deviations of the mean.
\\( z_{694} \\) is more than 3 standard deviations of the mean.

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x - \mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Identify values

We are given that $\mu = 690$, $\sigma=14$, and $x = 694$.

Step3: Calculate the z - score

Substitute the values into the z - score formula: $z=\frac{694 - 690}{14}=\frac{4}{14}\approx0.29$.

Step4: Analyze the z - score

A z - score of approximately 0.29 means that the value 694 is within 1 standard deviation (since 0.29 is between - 1 and 1) of the mean.

Answer:

$Z_{694}$ is within 1 standard deviation of the mean.