Pharmaceutical scientists tested the effectiveness of a new drug designed to cure malaria. They randomly assigned 400 infected lab rats to receive either the new drug or an existing drug and recorded how many rats were cured at the end of the study. Here are their results:
\begin{tabular}{lrrr}
& Cured & Not & Total \
\hline New drug & 176 & 24 & 200 \
Existing drug & 168 & 32 & 200
\end{tabular}

They want to use these results to construct a $99 \%$ confidence interval to estimate the difference in the proportion of these rats that would be cured under each drug (new - existing). Assume that all of the conditions for inference have been met.

Which of the following is a correct $99 \%$ confidence interval based on their data?

Choose 1 answer:
(A) $8 \pm 0.068$
(B) $8 \pm 0.089$
(C) $0.04 \pm 0.035$
(D) $0.04 \pm 0.068$
(ㄷ) $0.04 \pm 0.089$

Question

Pharmaceutical scientists tested the effectiveness of a new drug designed to cure malaria. They randomly assigned 400 infected lab rats to receive either the new drug or an existing drug and recorded how many rats were cured at the end of the study. Here are their results:
\begin{tabular}{lrrr} 
& Cured & Not & Total \
\hline New drug & 176 & 24 & 200 \
Existing drug & 168 & 32 & 200
\end{tabular}

They want to use these results to construct a $99 \%$ confidence interval to estimate the difference in the proportion of these rats that would be cured under each drug (new - existing). Assume that all of the conditions for inference have been met.

Which of the following is a correct $99 \%$ confidence interval based on their data?

Choose 1 answer:
(A) $8 \pm 0.068$
(B) $8 \pm 0.089$
(C) $0.04 \pm 0.035$
(D) $0.04 \pm 0.068$
(ㄷ) $0.04 \pm 0.089$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Calculate the sample proportions for each drug: $p_1 = \frac{176}{200}$ for the new drug and $p_2 = \frac{168}{200}$ for the existing drug. ‖ ‖ step 2 ⋮ Find the difference in sample proportions: $p_1 - p_2 = \frac{176}{200} - \frac{168}{200}$. ‖ ‖ step 3 ⋮ Calculate the standard error (SE) for the difference in proportions: $SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$. ‖ ‖ step 4 ⋮ Determine the z-score for a 99% confidence interval, which is approximately 2.576. ‖ ‖ step 5 ⋮ Calculate the margin of error (ME): $ME = z imes SE$. ‖ ‖ step 6 ⋮ Construct the 99% confidence interval for the difference in proportions: $(p_1 - p_2) \pm ME$. ‖ ***C*** ∻Key Concept∻ ⚹Confidence Interval for Difference in Proportions⚹ ∻Explanation∻ ⚹The confidence interval for the difference in proportions is calculated using the sample proportions, their standard error, and the appropriate z-score for the desired confidence level.⚹◊What formula would be used to calculate the confidence interval for the difference in proportions of rats cured between the new and existing drugs?⍭ Generate me a similar question◊

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