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, 460 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.

The 90% confidence interval is ( _____, _______ ). (Round to three decimal places as needed.

Answer 1

Solution :

Given that,

n = 900

x = 460

Point estimate = sample proportion = = x / n = 460/900=0.511

1 -   = 1- 0.511 =0.489

At 90% confidence level

= 1 - 90%  

= 1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z0.05 = 1.645 ( Using z table )

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

= 1.645 *((0.511*0.489) /900 )

E = 0.0274

A 90% confidence interval for population proportion p is ,

- E < p < + E

0.511-0.0274 < p < 0.511+0.0274

0.4836< p < 0.5384

The 90% confidence interval for the population proportion p is : 0.484, 0.538