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Mark noticed that the probability that a certain player hits a home run in a single game is $0.175$ . Mark is interested in the variability of the number of home runs if this player plays $200$ games. $\newline$ If Mark uses the normal approximation of the binomial distribution to model the number of home runs, what is the standard deviation for a total of $200$ games? Answer choices are rounded to the hundredths place. $\newline$ $0.14$ $\newline$ $28.88$ $\newline$ $5.92$ $\newline$ $5.37$

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Use normal distributions to approximate binomial distributions

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Q. Mark noticed that the probability that a certain player hits a home run in a single game is

0.175

. Mark is interested in the variability of the number of home runs if this player plays

200

games.

\newline

If Mark uses the normal approximation of the binomial distribution to model the number of home runs, what is the standard deviation for a total of

200

games? Answer choices are rounded to the hundredths place.

\newline

0.14

\newline

28.88

\newline

5.92

\newline

5.37

Identify given values and formula: Identify the given values and the formula to use for the standard deviation of a binomial distribution. $\newline$ The probability of hitting a home run in a single game $p$ is $0.175$ , and the number of games $n$ is $200$ . The standard deviation $\sigma$ of a binomial distribution is calculated using the formula: $\newline$ $\sigma = \sqrt{n \cdot p \cdot (1 - p)}$
Substitute values into formula: Substitute the given values into the standard deviation formula. $\sigma = \sqrt{200 \times 0.175 \times (1 - 0.175)}$
Calculate value inside square root: Calculate the value inside the square root. $\newline$ First, calculate $1 - p$ : $\newline$ $1 - 0.175 = 0.825$ $\newline$ Now, multiply $n$ , $p$ , and $(1 - p)$ together: $\newline$ $200 \times 0.175 \times 0.825 = 35 \times 0.825 = 28.875$
Find standard deviation: Take the square root of the value calculated in Step $3$ to find the standard deviation. $\newline$ $\sigma = \sqrt{28.875}$ $\newline$ $\sigma \approx 5.37$ (rounded to the hundredths place)

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