select the correct answer.\nin an experiment using 30 mice, the sample proportion of the mice that gained…

select the correct answer.\nin an experiment using 30 mice, the sample proportion of the mice that gained weight after a drug injection is 0.65. what is the 99.7% confidence interval for the actual proportion in the population?\n\na. between 0.602 and 0.697\nb. between 0.563 and 0.737\nc. between 0.476 and 0.824\nd. between 0.389 and 0.911

select the correct answer.\nin an experiment using 30 mice, the sample proportion of the mice that gained weight after a drug injection is 0.65. what is the 99.7% confidence interval for the actual proportion in the population?\n\na. between 0.602 and 0.697\nb. between 0.563 and 0.737\nc. between 0.476 and 0.824\nd. between 0.389 and 0.911

Answer

Explanation:

Step1: Identify the formula for confidence - interval of proportion

The formula for the confidence - interval of a proportion is $\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p}$ is the sample proportion, $n$ is the sample size, and $z$ is the z - score corresponding to the desired confidence level. For a 99.7% confidence level, the z - score $z = 3$. Here, $\hat{p}=0.65$ and $n = 30$.

Step2: Calculate the standard error

The standard error $SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Substitute $\hat{p}=0.65$ and $n = 30$ into the formula: [ \begin{align*} SE&=\sqrt{\frac{0.65\times(1 - 0.65)}{30}}\ &=\sqrt{\frac{0.65\times0.35}{30}}\ &=\sqrt{\frac{0.2275}{30}}\ &=\sqrt{0.007583}\ &\approx0.0871 \end{align*} ]

Step3: Calculate the margin of error

The margin of error $ME = z\times SE$. Since $z = 3$ and $SE\approx0.0871$, then $ME=3\times0.0871 = 0.2613$.

Step4: Calculate the confidence - interval

The lower limit of the confidence - interval is $\hat{p}-ME=0.65 - 0.2613=0.3887\approx0.389$. The upper limit of the confidence - interval is $\hat{p}+ME=0.65 + 0.2613=0.9113\approx0.911$.

Answer:

D. between 0.389 and 0.911