## Related questions with answers

The probability that a medical treatment is effective is 0.68, unknown to a researcher. In an experiment to investigate the effectiveness of the treatment, the researcher applies the treatment in 140 cases and measures whether the treatment is effective or not. What is the probability that the researcher’s estimate of the probability that the medical treatment is effective is within 0.05 of the correct answer?

Solution

VerifiedLet X has binomial distribution with parameters n and p. Then $\textbf{sample proportion}$ $\widehat{p}=\frac{X}{n}$ has normal distribution $N(p, \frac{p(1-p)}{n})$.

In our case p=0.68, n=140.

$\begin{align*} P(|\widehat{p}-p|\leq 0.05)&=P(-0.05\leq \frac{X}{140}-0.68 \leq 0.05)\\ &=P(0.63\leq \frac{X}{140}\leq 0.73)\\ &=P(0.63\cdot 140 \leq X \leq 0.73\cdot 140)\\ &=P(88.2 \leq X \leq 102.2)\\ &=P(89 \leq X \leq 102)\\ &\overset{X:B(140,0.68)}=\sum_{i=89}^{102}\binom{140}{i}0.68^i(1-0.68)^{140-i}\\ &=\fbox{0.795}. \end{align*}$

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