Question

The Harris Poll asked a sample of 1009 adults which causes of death they thought would become more common in the future. Gun violence topped the list: 706 members of the sample thought deaths from guns would increase. ?

(a) how to explain what the population proportion p is for this poll ?
(b) How to find a 95% confidence interval for p ?
(c) Harris announced a margin of error of plus or minus three percentage points for this poll result. Is there any effect on answer (b) ?
(d) If the confidence level is 90%, what is the difference between 95% and 90% confidence level ?

Answer 1

Answers bellow

Explanation:

a) the population proportion cannot be determined as that represents everyone. The sample, however, allows us to determine a point estimate, `hat p`, which we can assume is close the actual.

The Point estimate is,
`hat p= ("Everyone with a particular characteristic")/("Everyone in the sample")`

or in this case

`hat p= 706/1009`

b) to determine a confidence interval we use the equation,

`hat p-z*sigma < p < hat p+z*sigma`

where z is the z-score for your confidence level and `sigma` is your standard deviation.

For a 95% confidence z= 1.959963985 `~~`1.96

`sigma = sqrt((hat p (1-hatp))/n)`
`sigma = sqrt((706/1009 (1-706/1009))/1009)`

`sigma= 0.014431`

so subbing into,

`hat p-z*sigma < p < hat p+z*sigma`

`706/1009-1.96*0.014431 < p < 706/1009+1.96*0.014431`

`0.6714 < p < 0.728`

c) not sure exactly what you mean.
was think of using this equation but I'm not sure what you want to find out.

`n=(z^2p(1-p))/e^2` with e=0.03

d) a smaller confidence interval 95% `=>` 90% contains less values that the actual population could represent (remember that `hat p` is just an estimate of this value)

A 95% confidence interval means that 95% of sample means taken will lie within this range. A 90% confidence interval means that 90% will lie within its range, which is smaller. For example,

95% `=>` `0.6714 < p < 0.728`

90% `=>` `0.676 < p < 0.7234`

The 90% confidence interval is smaller and thus we are less confident that the actual population mean lies within its range.