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, 442 of the teenagers admitted using at least one informal element in school writing assignments. Based on the survey results, construct a 90% 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.

We are 90% confident that the true proportion of teenagers who have used at least one informal element in school writing assignments is between 0.464 and 0.518

Step-by-step explanation:

We can begin finding the estimator for the proportion of interest:

[tex]\hat p =\frac{442}{900}= 0.491[/tex]

The confidence level is 90% , and the significance level would be given by [tex]\alpha=1-0.90=0.1[/tex] and [tex]\alpha/2 =0.05[/tex]. And the critical value would be given by:

[tex]z_{\alpha/2}=-1.64, z_{1-\alpha/2}=1.64[/tex]

The confidence interval for the mean is given by the following formula:

[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

Replacing the info given we got:

[tex]0.491 - 1.64\sqrt{\frac{0.491(1-0.491)}{900}}=0.464[/tex]

[tex]0.491 + 1.64\sqrt{\frac{0.491(1-0.491)}{900}}=0.518[/tex]

We are 90% confident that the true proportion of teenagers who have used at least one informal element in school writing assignments is between 0.464 and 0.518