Question

Truthfulness in online profiles. 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

(a) Do these samples satisfy the guidelines for the large-sample confidence
interval?
(b) Give a 95% confidence interval for the difference between the proportions
of younger and older teens who include false information in their online
profiles.

Answer 1

(a) Do these samples satisfy the guidelines for the large-sample confidence interval?

Answer: Yes, these samples satisfy the guidelines for the large-sample confidence interval because both samples have sample sizes greater than 30, and both samples are random samples.

(b) Give a 95% confidence interval for the difference between the proportions
of younger and older teens who include false information in their online
profiles.

Confidence interval for difference between two population proportions:

Confidence interval = (P1 – P2) ± Z*sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]

Where, P1 and P2 are sample proportions for first and second groups respectively.

We are given

X1 = 117

N1 = 170

P1 = X1/N1 = 117/170 = 0.688235294

X2 = 152

N2 = 317

P2 = X2/N2 = 152/317 = 0.479495268

Confidence level = 95%

Critical Z value = 1.96

(by using z-table)

(P1 – P2) = 0.688235294 - 0.479495268 = 0.208740026

Standard error = sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]

Standard error = sqrt[(0.688235294*(1 – 0.688235294)/170) + (0.479495268*(1 – 0.479495268)/317)]

Standard error = 0.0453

Confidence interval = (P1 – P2) ± Z*sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]

Confidence interval = (P1 – P2) ± Z*sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]

Confidence interval = 0.208740026 ± 1.96*0.0453

Confidence interval = 0.208740026 ± 0.0887

Lower limit = 0.208740026 - 0.0887 = 0.1200

Upper limit = 0.208740026 + 0.0887 = 0.2975

Confidence interval = (0.1200, 0.2975)