Question

Many teens have posted profiles on a social-networking website. A sample survey in 2007 asked a random sample of teens with online profiles if they included false information in their profiles. Of 170 younger teens (aged 12 to 14), 117 said “yes.” Of 317 older teens (aged 15 to 17), 152 said “yes.”

(16 Points) a. Does the proportion of younger teens (aged 12-14) who include false information on their profiles differ from that of older teens (aged 15-17)? Test at the 5% significance level.

(5 Points) b. Calculate a 95% confidence interval for the difference between proportions of younger teens (aged 12-14) and older teens (aged 15-17) who included false information on their profiles. You may refer to assumptions met in part a. Show your work.

(4 Points) c. Interpret the confidence interval in the context of the problem.

Answer 1

a) The null hypothesis:

The alternate hypothesis:  

where,   is the proportion of younger teens who include false information = 117/170 = 0.69

is the proportion of older teens who include false information = 152/317 = 0.48

Test statistic:

where,   and  

  

Here, the level of significance:

Since, this is a two sided test, we calculate  

Therefore, we reject the null hypothesis at 5% level of significance and conclude that the proportion of younger teens (age 12 to 14) who include false information on their profile differ from that of the older teens ( age 15 to 17).

b) The 95% confidence interval for the difference is:

c) Interpretation: The 95% confidence interval is (0.120,0.297) i.e, we are 95% confident that the true difference between the population proportion is in the range defined by  . Since both the end of the confidence interval are positive we can conclude that more younger teens include false information on their profiles than older kids.