Question

a supervisor finds the mean number of miles that the employees in a department live from work. he finds \\( \bar{x} = 29 \\) and \\( s = 3.8 \\). which statement must be true?\
\\( z_{37} \\) is within 1 standard deviation of the mean.\
\\( z_{37} \\) is between 1 and 2 standard deviations of the mean.\
\\( z_{37} \\) is between 2 and 3 standard deviations of the mean.\
\\( z_{37} \\) is more than 3 standard deviations of the mean.

Explanation:

Step1: Recall z-score formula

The z - score formula is $z=\frac{x - \overline{x}}{s}$, where $x$ is the value, $\overline{x}$ is the mean, and $s$ is the standard deviation.

Step2: Identify values

Here, $x = 37$, $\overline{x}=29$, and $s = 3.8$.

Step3: Calculate z - score

Substitute the values into the formula: $z=\frac{37 - 29}{3.8}=\frac{8}{3.8}\approx2.11$.

Step4: Analyze z - score range

A z - score of approximately 2.11 is between 2 and 3? Wait, no. Wait, 2 < 2.11 < 3? Wait, 2.11 is between 2 and 3? Wait, no, 2.11 is between 2 and 3? Wait, 1 standard deviation from the mean is $\overline{x}\pm s=29\pm3.8=(25.2,32.8)$. 2 standard deviations is $\overline{x}\pm2s = 29\pm7.6=(21.4,36.6)$. 3 standard deviations is $\overline{x}\pm3s=29\pm11.4=(17.6,40.4)$. Now, 37: 36.6 is the upper bound of 2 standard deviations, and 40.4 is the upper bound of 3 standard deviations. So 37 is between $2s$ and $3s$ above the mean. So the z - score for 37 is $\frac{37 - 29}{3.8}=\frac{8}{3.8}\approx2.11$, which is between 2 and 3 standard deviations? Wait, no, 2 < 2.11 < 3? Wait, 2.11 is between 2 and 3. Wait, the options: "between 2 and 3 standard deviations" is one of the options. Wait, let's recalculate: $37 - 29 = 8$. $8\div3.8\approx2.105$. So the z - score is approximately 2.11, which is between 2 and 3? Wait, no, 2 < 2.11 < 3? Yes, because 2.11 is greater than 2 and less than 3. Wait, but let's check the intervals:

• 1 standard deviation: $29 - 3.8 = 25.2$ to $29+3.8 = 32.8$
• 2 standard deviations: $29 - 7.6 = 21.4$ to $29 + 7.6 = 36.6$
• 3 standard deviations: $29 - 11.4 = 17.6$ to $29+11.4 = 40.4$

37 is greater than 36.6 (the upper bound of 2 standard deviations) and less than 40.4 (the upper bound of 3 standard deviations). So 37 is between 2 and 3 standard deviations from the mean. So the z - score is between 2 and 3. So the correct option is "Z₃₇ is between 2 and 3 standard deviations of the mean."

Answer:

Z₃₇ is between 2 and 3 standard deviations of the mean. (The option with this statement)