Question

An article included the following statement: "Few people believe there's much reality in reality TV: a total of 78% said the shows are either 'totally made up' or 'mostly distorted.'" This statement was based on a survey of 1006 randomly selected adults. Compute a bound on the error (based on 95% confidence) of estimation for the reported proportion of 0.78. (Round your answer to three decimal places.)


Interpret the bound. (Round your answers to one decimal place.)

We are % confident that the proportion of all adults who believe that the shows are either "totally made up" or "mostly distorted" is within % of the sample proportion of %.

Answer 1

Solution :

Given that,

n = 1006

Point estimate = sample proportion = =0.78

1 -   = 1-0.78 =0.22

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.96   ( Using z table )

Margin of error = E = Z/2   * ((( * (1 - )) / n)

= 1.96 (((0.78*0.22) /1006 )

E = 0.026

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.78- 0.026< p < 0.78+0.026

0.754< p < 0.806

We are 99% confident that the proportion of all adults who believe that the shows are either "totally made up" or "mostly distorted" is within 7.5% of the sample proportion of 8.1%.