Consider a random sample of 25 observations which follow a normal distribution N(0,1). Which of the following is an approximate 95% probability interval for the sample proportion of observations larger than 1?
(0.06, 0.60)
(0.01, 0.15)
(0.03, 0.20)
(0.02, 0.30)
(0.15, 0.60)
I know that I have to find the probability of success which is binomial and then use the distribution of p-hat to find the 95% Probability interval. I'm just having problems on finding the probability of success (P(X>1))
would really appreciate your help!! :)
We must first find the probability that an individual observation is larger than one. Letting $X\sim \mathsf N(0,1)$ we get
$$P(X\gt1)=1-P(X<1)=1-\Phi\left(1\right)\approx0.1587$$
A $95$% confidence interval for $p$, the sample proportion, is given by
$$p_0\pm 1.96\cdot\sqrt{\frac{p_0(1-p_0)}{n}}$$
where $p_0\approx0.1587$ and $n=25$