What is the sampling distribution of the proportion of adults who smoke out of a random sample of 300?
In a study where 300 randomly selected adults were asked whether they smoked, 22.4% were found to be smokers. Using this information, give an estimate for the probability that in another random sample of 300, 50 will be found to smoke.
The sampling distribution of the number out of a random sample of 300 adults who smoke, `x`, can be assumed to be Binomial(300,` ``p`) where `p` is the true proportion who smoke in the (adult) population as a whole. The sampling distribution of the proportion in the sample who smoke `hat(p) = x/300` can be assumed to be Binomial(300,`p`)/300
With an estimate for `p` of `hat(p)=0.224` we can estimate the distribution of the sample count (over repeated samples) to be Binomial(300,`hat(p)=0.224`).
Using this estimated distribution, we can estimate the probability that 50 adults out of a random sample of 300 will smoke. This is
`(300!)/(50!250!)hat(p)^50(1-hat(p))^250 = 0.0087`
The sampling distribution of `hat(p)` can be assumed to be Binomial(300,p)/300.
The probability that 50 adults will be found to be smokers in a random sample of 300 is estimated as 0.0087.