miss gruber ran a sit - up competition among her p.e. students and monitored how many sit - ups each student could do.
x is the number of sit - ups that a randomly chosen student did. what is the standard deviation of x?
round your answer to the nearest hundredth.

Question

Accepted Answer

# Explanation:
## Step1: Calculate the total number of students
The total number of students $n=1 + 3+2 + 5+1+8=20$
## Step2: Calculate the expected value $E(X)$
$$
\begin{align*}
E(X)&=\frac{67\times1 + 68\times3+101\times2 + 122\times5+164\times1+177\times8}{20}\
&=\frac{67+204 + 202+610+164+1416}{20}\
&=\frac{2663}{20}=133.15
\end{align*}
$$
## Step3: Calculate $E(X^{2})$
$$
\begin{align*}
E(X^{2})&=\frac{67^{2}\times1+68^{2}\times3 + 101^{2}\times2+122^{2}\times5+164^{2}\times1+177^{2}\times8}{20}\
&=\frac{4489\times1 + 4624\times3+10201\times2+14884\times5+26896\times1+31329\times8}{20}\
&=\frac{4489+13872+20402+74420+26896+250632}{20}\
&=\frac{390711}{20}=19535.55
\end{align*}
$$
## Step4: Calculate the variance $V(X)$
The variance $V(X)=E(X^{2})-(E(X))^{2}$
$$
\begin{align*}
V(X)&=19535.55-(133.15)^{2}\
&=19535.55 - 17729.9225\
&=1805.6275
\end{align*}
$$
## Step5: Calculate the standard deviation $\sigma$
The standard deviation $\sigma=\sqrt{V(X)}$
$\sigma=\sqrt{1805.6275}\approx  42.50$

# Answer:
$42.50$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the total number of students

Step2: Calculate the expected value \(E(X)\)

Step3: Calculate \(E(X^{2})\)

Step4: Calculate the variance \(V(X)\)

Step5: Calculate the standard deviation \(\sigma\)

Answer: