Question

Candidate A is facing two opposing candidates in a mayoral election. In a recent poll of 300 residents, 98 supported candidate B and 58 supported candidate C. Construct a 95% confidence interval on the population proportion for the support of candidate A in the following election. [0.4781, 0.4819] [0.4235, 0.5365] [0.3942, 0.5658] [0.4057, 0.5543]

Answer 1

Answer:

[0.4235, 0.5365]

Step-by-step explanation:

Data given and notation  

n=300 represent the random sample taken    

X=300-98-58=144 represent the people that support the candidate A in the sample

$\hat p=\frac{144}{300}=0.48$ estimated proportion of people that support the candidate A in the sample

$\alpha=0.05$ represent the significance level

Confidence =0.95 or 95%

p= population proportion of people that support the candidate A.

Confidence interval

The confidence interval would be given by this formula

$\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}$

For the 95% confidence interval the value of $\alpha=1-0.95=0.05$ and $\alpha/2=0.025$, with that value we can find the quantile required for the interval in the normal standard distribution.

$z_{\alpha/2}=1.96$

And replacing into the confidence interval formula we got:

$0.48 - 1.96 \sqrt{\frac{0.48(1-0.48)}{300}}=0.4235$

$0.48 + 1.96 \sqrt{\frac{0.48(1-0.48)}{300}}=0.5365$

And the 95% confidence interval would be given (0.4235;0.5365).

$0.4235 \leq p \leq 0.5365$