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 ?

Oct 24, 2016

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,
$\hat{p} = \left(\text{Everyone with a particular characteristic")/("Everyone in the sample}\right)$

or in this case

$\hat{p} = \frac{706}{1009}$

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

$\hat{p} - z \cdot \sigma < p < \hat{p} + z \cdot \sigma$

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

For a 95% confidence z= 1.959963985 $\approx$1.96

$\sigma = \sqrt{\frac{\hat{p} \left(1 - \hat{p}\right)}{n}}$
$\sigma = \sqrt{\frac{\frac{706}{1009} \left(1 - \frac{706}{1009}\right)}{1009}}$

$\sigma = 0.014431$

so subbing into,

$\hat{p} - z \cdot \sigma < p < \hat{p} + z \cdot \sigma$

$\frac{706}{1009} - 1.96 \cdot 0.014431 < p < \frac{706}{1009} + 1.96 \cdot 0.014431$

$0.6714 < p < 0.728$

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

$n = \frac{{z}^{2} p \left(1 - p\right)}{e} ^ 2$ with e=0.03

d) a smaller confidence interval 95% $\implies$ 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% $\implies$$0.6714 < p < 0.728$

90% $\implies$$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.