Suppose that Y has a Binomial distribution with n=80 and p=0.67. Let widehat(p)=Y//80. The standard deviation of the distribution of hat(p) is ____. (A) (17.688)/(80) (B) 0.67 (C) 53.6 (D) sqrt((0.22 bar(11))/(80))

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Given formula for standard deviation: We know that $\hat{p}$ is the sample proportion, which is the estimator of the population proportion $p$ in a binomial distribution. The standard deviation of the sample proportion $\hat{p}$ is given by the formula: $$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$ where $p$ is the probability of success, $1-p$ is the probability of failure, and $n$ is the sample size.Substitute values into formula: Given that $n=80$ and $p=0.67$, we can substitute these values into the formula to calculate the standard deviation of $\hat{p}$: $$\sigma_{\hat{p}} = \sqrt{\frac{0.67(1-0.67)}{80}}$$Calculate 1-p: First, calculate $1-p$: $$1-0.67 = 0.33$$Multiply p by 1-p: Now, multiply $p$ by $1-p$: $$0.67 \times 0.33 = 0.2211$$Divide product by sample size: Next, divide this product by the sample size $n$: $$\frac{0.2211}{80} = 0.00276375$$Find square root of result: Finally, take the square root of the result to find the standard deviation of $\hat{p}$: $$\sigma_{\hat{p}} = \sqrt{0.00276375} \approx 0.05257$$Check closest option: The final answer needs to be in the form of one of the given options. The calculated standard deviation is approximately 0.05257, which is not exactly in the form of any of the options provided. However, we can recognize that the square root of a fraction is involved, and the closest option that represents this is option (D), which is in the form of a square root of a fraction over 80. Let's express our calculated standard deviation in a similar form to see if it matches: $$\sigma_{\hat{p}} = \sqrt{\frac{0.2211}{80}}$$Confirm equivalence with 2/9: We can see that the fraction inside the square root is slightly different from the one in option (D). The fraction in option (D) is $\frac{0.22\overline{11}}{80}$, which is a repeating decimal equivalent to $\frac{2}{9}$. Let's check if our calculated fraction is equivalent to $\frac{2}{9}$: $$0.2211 \approx \frac{2}{9}$$Conclusion: To confirm, we can divide 2 by 9: $$\frac{2}{9} = 0.2222\overline{2}$$ This is very close to our calculated value of 0.2211, and the slight difference is likely due to rounding during the calculation. Therefore, we can conclude that the standard deviation of $\hat{p}$ is indeed represented by option (D): $$\sigma_{\hat{p}} = \sqrt{\frac{0.22\overline{11}}{80}}$$

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