Question

Math

Posted 3 months ago

```
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) $0.04 \pm 0.089$
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

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): $ME = z \times SE$

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.

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