Gloria enjoys playing Skee ball - an arcade game where players toss a ball so it rolls up a ramp and falls into one of several slots. Each slot is worth a different number of points based on how difficult it is to land the ball in that slot. Gloria only aims at the most difficult slot, which is worth 100 points, but she only has a $10 \%$ chance of landing the ball in that slot. If she misses, her ball will certainly land in the 10 point slot.

The table below displays the probability distribution of $X$, the number of points Gloria scores on a random shot.
\begin{tabular}{lll}
\hline$X=$ points & 100 & 10 \
$P(X)$ & $10 \%$ & $90 \%$
\end{tabular}

Given that $\mu_{X}=19$ points, calculate $\sigma_{X}$.
$\sigma_{X}=$ $\square$ points

Question

Gloria enjoys playing Skee ball - an arcade game where players toss a ball so it rolls up a ramp and falls into one of several slots. Each slot is worth a different number of points based on how difficult it is to land the ball in that slot. Gloria only aims at the most difficult slot, which is worth 100 points, but she only has a $10 \%$ chance of landing the ball in that slot. If she misses, her ball will certainly land in the 10 point slot.

The table below displays the probability distribution of $X$, the number of points Gloria scores on a random shot.
\begin{tabular}{lll}
\hline$X=$ points & 100 & 10 \
$P(X)$ & $10 \%$ & $90 \%$
\end{tabular}

Given that $\mu_{X}=19$ points, calculate $\sigma_{X}$.
$\sigma_{X}=$ $\square$ points

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the standard deviation $\sigma_X$, we first need to calculate the variance $\sigma_X^2$. The variance is given by the formula $\sigma_X^2 = E[X^2] - (\mu_X)^2$, where $E[X^2]$ is the expected value of the square of X. ‖ ‖ step 2 ⋮ Calculate $E[X^2]$ using the probability distribution. $E[X^2] = (100^2 imes 0.10) + (10^2 imes 0.90)$. ‖ ‖ step 3 ⋮ Simplify the calculation of $E[X^2]$. $E[X^2] = (10000 imes 0.10) + (100 imes 0.90) = 1000 + 90 = 1090$. ‖ ‖ step 4 ⋮ Use the given mean $\mu_X = 19$ to calculate the variance. $\sigma_X^2 = E[X^2] - (\mu_X)^2 = 1090 - (19)^2$. ‖ ‖ step 5 ⋮ Simplify the calculation of the variance. $\sigma_X^2 = 1090 - 361 = 729$. ‖ ‖ step 6 ⋮ Calculate the standard deviation $\sigma_X$ by taking the square root of the variance. $\sigma_X = \sqrt{729}$. ‖ ‖ step 7 ⋮ Simplify the calculation of the standard deviation. $\sigma_X = 27$. ‖ ∻Answer∻ ⚹ $\sigma_{X}=27$ points ⚹ ∻Key Concept∻ ⚹Standard Deviation and Variance of a Probability Distribution⚹ ∻Explanation∻ ⚹The standard deviation is the square root of the variance, which measures the spread of a probability distribution around the mean. It is calculated using the expected value of the square of the random variable and the square of the mean.⚹◊What is the formula to calculate the standard deviation of a probability distribution⍭ Generate me a similar question◊

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