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Approximately 20% of newborns are born more than 1 week before their due date. A random sample of 20 newborns is selected.
The standard deviation of the sampling distribution for the proportion of your sample that is born more than 7 days before their due date is
(A) 0.20
(B) 3.2
(C) 0.089
(D) 0.008

Approximately $20\%$ of newborns are born more than $1$ week before their due date. A random sample of $20$ newborns is selected. $\newline$ The standard deviation of the sampling distribution for the proportion of your sample that is born more than $7$ days before their due date is $\newline$ (A) $0.20$ $\newline$ (B) $3.2$ $\newline$ (C) $0.089$ $\newline$ (D) $0.008$

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Find probabilities using the binomial distribution

Full solution

Q. Approximately

20\%

of newborns are born more than

1

week before their due date. A random sample of

20

newborns is selected.

\newline

The standard deviation of the sampling distribution for the proportion of your sample that is born more than

7

days before their due date is

\newline

(A)

0.20

\newline

(B)

3.2

\newline

(C)

0.089

\newline

(D)

0.008

Identify Proportion: First, we need to identify the proportion of newborns born more than $1$ week early, which is $20\%$ or $0.20$ .
Calculate Standard Deviation: Next, we calculate the standard deviation of the sampling distribution for the proportion using the formula for the standard deviation of a sample proportion: $\newline$ Standard deviation = $\sqrt{\left(p \times (1 - p)\right) / n}$ $\newline$ where $p$ is the proportion and $n$ is the sample size.
Plug in Values: Plug in the values: $p = 0.20$ and $n = 20$ . $\newline$ Standard deviation = $\sqrt{\left(0.20 \times (1 - 0.20)\right) / 20}$
Do the Calculation: Do the calculation: $\newline$ Standard deviation = $\sqrt{\left(0.20 \times 0.80\right) / 20}$ $\newline$ Standard deviation = $\sqrt{0.16 / 20}$ $\newline$ Standard deviation = $\sqrt{0.008}$
Calculate Square Root: Finally, calculate the square root of $0.008$ . Standard deviation = $\sqrt{0.008} \approx 0.089$ (rounded to three decimal places).

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