Jerry took a sample of 4 employees in his office and observed how many hours they each worked one day. Here is what he found:
\begin{tabular}{lrrrrr}
Employee & Leslie & April & Tom & Andy \
\cline { 1 - 2 } \cline { 3 - 4 } & 10 & 2 & 4 & 8
\end{tabular}

Jerry found their mean was $\bar{x}=6$ hours. He thinks the standard deviation is
$$
s_{x}=\sqrt{\frac{(10-6)^{2}+(2-6)^{2}+(4-6)^{2}+(8-6)^{2}}{5}}
$$

What is the error in Jerry's standard deviation calculation?
Choose 1 answer:
(A) He shouldn't take the square root at all.
(B) He should only take the square root of the numerator.
(c) His denominator is incorrect.
(D) He shouldn't square each deviation in the numerator.
(E) There is no error; his calculation is correct.

Question

Jerry took a sample of 4 employees in his office and observed how many hours they each worked one day. Here is what he found:
\begin{tabular}{lrrrrr} 
Employee & Leslie & April & Tom & Andy \
\cline { 1 - 2 } \cline { 3 - 4 } & 10 & 2 & 4 & 8
\end{tabular}

Jerry found their mean was $\bar{x}=6$ hours. He thinks the standard deviation is
$$
s_{x}=\sqrt{\frac{(10-6)^{2}+(2-6)^{2}+(4-6)^{2}+(8-6)^{2}}{5}}
$$

What is the error in Jerry's standard deviation calculation?
Choose 1 answer:
(A) He shouldn't take the square root at all.
(B) He should only take the square root of the numerator.
(c) His denominator is incorrect.
(D) He shouldn't square each deviation in the numerator.
(E) There is no error; his calculation is correct.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the correct formula for the sample standard deviation, which is $s_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$, where $x_i$ are the sample values, $\bar{x}$ is the sample mean, and $n$ is the sample size. ‖ ‖ step 2 ⋮ Note that Jerry used the denominator of 5 instead of the correct denominator $n-1$, which should be $4-1=3$ for a sample size of 4. ‖ ‖ step 3 ⋮ Correct Jerry's calculation by using the right denominator: $s_x = \sqrt{\frac{(10-6)^2 + (2-6)^2 + (4-6)^2 + (8-6)^2}{3}}$. ‖ ∻[question number] Answer∻ ***C*** ∻Key Concept∻ ⚹Sample Standard Deviation⚹ ∻Explanation∻ ⚹The sample standard deviation formula requires dividing by $n-1$ instead of $n$ to account for the degrees of freedom in the sample. Jerry used the wrong denominator in his calculation.⚹◊What is the correct formula for calculating the standard deviation of a sample?⍭ Generate me a similar question◊

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