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.
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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.
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