Question

Due to the popularity of instant messaging and social​ networking, informal elements such as emoticons​ (e.g., the symbol​ ":)" to represent a​ smile) and abbreviations​ (e.g., "LOL" for​ "laughing out​ loud") have worked their way into​ teenagers' school writing assignments. A survey interviewed 900 randomly selected teenagers by telephone on their writing habits.​ Overall, 422 of the teenagers admitted using at least one informal element in school writing assignments. Based on the survey​ results, construct a 99​% confidence interval for the proportion of all teenagers who have used at least one informal element in school writing assignments. Give a practical interpretation of the interval.

The 99% confidence interval is ( _____, _______ ). (Round to three decimal places as needed.)

Answer 1

Solution :

Given that,

n = 900

x = 422

Point estimate = sample proportion = = x / n = 422 / 900 = 0.469

1 - = 1 - 0.469 = 0.531

At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

Z/2 = Z0.005 = 2.576

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

= 2.576 (((0.469 * 0.531) / 900)

= 0.043

A 99% confidence interval for population proportion p is ,

± E   

= 0.469  ± 0.043

= ( 0.426, 0.512 )

We are 99% confident that the true proportion of  all teenagers who have used at least one informal element in school writing assignments between 0.426 and 0.512.