Question

A doctor treats 65 brain cancer cases, and he found 15 of the cases survived at least 3 years.
a) Suppose the doctor wants to construct a confidence interval for the proportion of all the patients who have brain cancer and survived at least 3 years, what is the preliminary conditions that need to be verified and what can we assume once the conditions are satisfied?
b) Construct a 86% confidence interval for the proportion of all patients who have brain cancer and survived at least 3 years, and state the confidence statement.

Answer 1

a) we first check for randomness and independence. Once the conditions are satisfied we assume normality.

b) 86% CI=$\hat p±z^*\sqrt{\frac{\hat p(1-\hat p)}{n}}$

$\hat p=\frac {15}{65}=0.2308$

=$0.2308±1.48\sqrt{\frac{0.2308*(1-0.2308)}{65}}$

=(0. 1534,0.3081)

We are 86% confident that the population proportion value is within the range 0.1534 and 0.3081.