In a sample of 1000 randomly selected consumers who had opportunities to send in a rebate claim form after purchasing a product, 260 of these people said they never did so. Reasons cited for their behavior included too many steps in the process, amount too small, missed deadline, fear of being placed on a mailing list, lost receipt, and doubts about receiving the money. Calculate an upper confidence bound at the 95% confidence level for the true proportion of such consumers who never apply for a rebate. (Round your answer to four decimal places.)

Given : In a sample of 1000 randomly selected consumers who had opportunities to send in a rebate claim form after purchasing a product, 260 of these people said they never did so.

i.e. n= 1000 and x= 260

⇒ Sample proportion : [tex]\hat{p}=\dfrac{260}{1000}=0.26[/tex]

z-value for 95% confidence interval : [tex]z_c=1.960[/tex]

Now, an upper confidence bound at the 95% confidence level for the true proportion of such consumers who never apply for a rebate. :-

[tex]\hat{p}+ z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

[tex]=0.26+ (1.96)\sqrt{\dfrac{0.26(1-0.26)}{1000}}[/tex]

[tex]=0.26+ (1.96)(0.01387)=0.26+0.0271852=0.2871852\approx0.2872[/tex]

∴ An upper confidence bound at the 95% confidence level for the true proportion of such consumers who never apply for a rebate : 0.2872