An egg supplier to a restaurant sends a daily shipment of 500 eggs. Previous records show that $4 \%$ of these eggs arrive damaged to the restaurant. Let $X=$ the number of eggs that arrive damaged on a randomly selected day.

Find the mean and standard deviation of $X$.
You may round your answers to the nearest tenth.
$$
\begin{array}{l}
\mu_{X}=\square \text { eggs } \
\sigma_{X} \approx \square \text { eggs }
\end{array}
$$
eggs eggs

Question

An egg supplier to a restaurant sends a daily shipment of 500 eggs. Previous records show that $4 \%$ of these eggs arrive damaged to the restaurant. Let $X=$ the number of eggs that arrive damaged on a randomly selected day.

Find the mean and standard deviation of $X$.
You may round your answers to the nearest tenth.
$$
\begin{array}{l}
\mu_{X}=\square \text { eggs } \
\sigma_{X} \approx \square \text { eggs }
\end{array}
$$
eggs eggs

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the mean $\mu_X$ of the number of damaged eggs, we use the probability of an egg being damaged and the total number of eggs. ‖ ‖ step 2 ⋮ The probability of an egg being damaged is $4\%$, which is $0.04$ as a decimal. ‖ ‖ step 3 ⋮ The total number of eggs sent daily is $500$. ‖ ‖ step 4 ⋮ The mean $\mu_X$ is calculated by multiplying the total number of eggs by the probability of an egg being damaged: $\mu_X = 500 imes 0.04$. ‖ ‖ step 5 ⋮ Calculating the mean: $\mu_X = 500 imes 0.04 = 20$ eggs. ‖ ‖ step 6 ⋮ To find the standard deviation $\sigma_X$, we use the formula for the standard deviation of a binomial distribution, which is $\sigma = \sqrt{n \cdot p \cdot (1-p)}$, where $n$ is the number of trials, and $p$ is the probability of success. ‖ ‖ step 7 ⋮ Substituting $n = 500$ and $p = 0.04$ into the formula: $\sigma_X = \sqrt{500 imes 0.04 imes (1 - 0.04)}$. ‖ ‖ step 8 ⋮ Calculating the standard deviation: $\sigma_X = \sqrt{500 imes 0.04 imes 0.96} \approx \sqrt{19.2} \approx 4.4$ eggs. ‖ ∻[1] Answer∻ ⚹ $\mu_X = 20$ eggs ⚹ ⚹ $\sigma_X \approx 4.4$ eggs ⚹ ∻Key Concept∻ ⚹Binomial Distribution⚹ ∻Explanation∻ ⚹The mean and standard deviation of a binomial distribution with $n$ trials and probability $p$ of success are calculated using the formulas $\mu = n \cdot p$ and $\sigma = \sqrt{n \cdot p \cdot (1-p)}$.⚹◊What is the formula to calculate the standard deviation of a binomial distribution?⍭ Generate me a similar question◊

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