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) 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.