Question

Many businesses use direct mail (oftentimes called "junk mail") to advertise their products and services, and with the declining volume of physical mail, the United States Postal Service has encouraged its use in the last few years. Direct mail advertising involves sending solicitations (credit card promotions, special deals at car dealerships and furniture stores, grocery store circulars, etc.) to many thousands of people in hopes that a proportion of those people will end up buying products or services from them. These businesses purchase consumer information data sets from "data brokers," that, in turn, gather and consolidate consumer data from various sources such as warranty cards, surveys, government and public records, social media sites such as Linkedin and Facebook, and other places. In case you are interested, you can find more information here: https://www.privacyrights.org/content/data-brokers-and-your-privacy, although you do not need to do this to answer the question.

Suppose a company decides to test out a new layout on a flier before fully committing to a nationwide mailing campaign. It mails the flier to a sample of 1,259 people randomly selected from a mailing list data base. The company gets a response from 117 of the recipients. Find the upper bound of a 95% confidence interval for the true proportion of people nationwide who would respond to the ad. Round your answer to three decimalplaces.

Answer 1

Solution :

Given that,

n = 1259

x = 117

Point estimate = sample proportion = $\hat p$ = x / n = 117 /1259 = 0.093

1 - $\hat p$ = 1-0.093 = 0.907

At 95% confidence level

$\alpha$ = 1-0.95% =1-0.95 =0.05

$\alpha$/2 =0.05/ 2= 0.025

Z$\alpha$/2 = Z_{0.025 = 1.960}

Z_$\alpha$/2 = 1.960

Margin of error = E = Z_{$\alpha$ / 2} * $\sqrt$ (($\hat p$ * (1 - $\hat p$ )) / n)

= 1.960 * ($\sqrt$(0.093*(0.907) /1259 )

= 0.016

A 95% confidence interval for population proportion p is ,

$\hat p$ - E < p < $\hat p$ + E

0.093 - 0.016 < p < 0.093+ 0.016

0.077 < p < 0.109  

(0.077 ,0.109 )