The people at the Cloak soft drink company believe that their brand of drink is favored over their competitor's brand, Pempsi, by the majority of people. A quick survey is conducted to test this belief. Thirty people are asked whether they prefer Cloak or Pempsi, and 21 people did prefer Cloak.

A statistician at Cloak wishes to construct a 95% confidence interval for the proportion of people that prefer Cloak soft drink.

Find lower bound and upper bound of confidance interval.

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

X=21 represent the people that prefer Cloak

[tex]\hat p=\frac{21}{30}=0.7[/tex] estimation for the sample proportion

n=30 sample size selected

Confidence =0.95 or 95%

The population proportion have the following distribution

[tex]p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})[/tex]

The confidence interval would be given by this formula

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

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

[tex]z_{\alpha/2}=1.96[/tex]

And replacing into the confidence interval formula we got:

[tex]0.7 - 1.96 \sqrt{\frac{0.7(1-0.7)}{30}}=0.536[/tex]

[tex]0.7 + 1.96 \sqrt{\frac{0.7(1-0.7)}{30}}=0.864[/tex]

And the 95% confidence interval would be given (0.536;0.894).