Question

A researcher studying public opinion of proposed Social Security changes obtains a simple random sample of 35 adult Americans and asks them whether or not they support the proposed changes. To say that the distribution of the sample proportion of adults who respond​ yes, is approximately​ normal, how many more adult Americans does the researcher need to sample in the following​ cases?

Answer 1

Answer: a) 15, b) 1.

Explanation:

A researcher studying public opinion of proposed social security changes obtains a simple random sample of 35 adult Americans and asks them whether or not they support the proposed changes. To say that the distribution of the sample proportion of adults who respond yes, is approximately normal, how many more Americans does the researcher need to sample in the following cases?

(a) 10% of all adult Americans support the changes

Here,
`np>5\ and\ n(1-p)>5`

And ,
`p=0.10`

So, consider the equality, to find the value of 'n'.

`np=5\\\\n* 0.10=5\\\\n=(5)/(0.10)\\\\n=50`

So, there are
`50-35=15` more adult Americans needed.

(b) 15% of all adult Americans supports the changes

Here,
`p=0.15`

So, again we get that

`np=5\\\\n* 0.15=5\\\\n=(5)/(0.15)\\\\n=33.33\\\\n\approx 34`

So, there are
`35-34=1` more adult Americans needed.

Hence, a) 15, b) 1.