# According to the National Center for Health Statistics (2004), 22.4% of adults in the US are smokers. A random sample of 300 adults is obtained. a. Describe the sampling distribution of `hat(p)`,...

According to the National Center for Health Statistics (2004), 22.4% of adults in the US are smokers. A random sample of 300 adults is obtained.

a. Describe the sampling distribution of `hat(p)`, the sample proportion of adults who smoke.

b. In a random sample of 300 adults, what is the probability that at least 50 are smokers?

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### 1 Answer

Assuming the quoted population statistic is correct, the sampling distribution of `X = 300hat(p)`, the *number* who smoke in a random sample of 300 adults, can be assumed to be Binomial(300,`p`= 0.224). The sampling distribution of `hat(p) = X/300`, the *proportion *who smoke in a random sample of 300 adults, can then be assumed to be scaled Binomial(300,`p`= 0.224) where the scale factor is 300.

Given that the sampling distribution of the count in a sample is assumed to be Binomial(300,0.224), the probability that at least 50 adults in a random sample of 300 will be found to be smokers is (using lookup tables)

`Pr(X>50) = 0.991`

**a) The sampling distribution is scaled Binomial(300,0.224) where the scale factor is 300**.

**b) The probability that a random sample of 300 adults will contain at least 50 smokers is 0.991.**