Question

A research center project involved a survey of 851 Internet users. It provided a variety of statistics on Internet users. (a) The sample survey showed that 92% of respondents said the Internet has been a good thing for them personally. Develop a 95% confidence interval for the proportion of respondents who say the Internet has been a good thing for them personally. (Round your answers to four decimal places.)

Answer 1

The answer is "(0.9193924 , 0.9206076)".

Explanation:

`\text{sample proportion}\ (SP) = 0.92\\\\\text{sample size}\ n = 851\\\\\text{Standard error} \ SE = \sqrt{((SP *(1 - SP))/(n))}\\\\`

`= \sqrt{((0.92 * (0.08))/(851))}\\\\= \sqrt{(0.0736)/(851)}\\\\= \sqrt{8.648* 10^(-5)}\\\\=0.00031`

`\text{CI level is}\ 95\% \\\\\therefore\\\\ \alpha = 1 - 0.95 = 0.05\\\\(\alpha)/(2) = (0.05)/(2) = 0.025\\\\ Z_c = Z_{((\alpha)/(2))} = 1.96`

Calculating the Margin of Error:

`ME = z_(c) * SE\\\\`

`= 1.96 * 0.00031\\\\ = 0.0006076`

`CI = (SP - z*SE, SP + z*SE)`

`= (0.92 - 1.96 * 0.00031 , 0.92 + 1.96 * 0.00031)\\\\ = (0.92 - 0.0006076 , 0.92 + 0.0006076)\\\\= (0.9193924 , 0.9206076)`