Question

A survey of 270 young professionals found that one dash eighth of them use their cell phones primarily for​ e-mail. Can you conclude statistically that the population proportion who use cell phones primarily for​ e-mail is less than 0.17​? Use a​ 95% confidence interval. The​ 95% confidence interval is [nothing comma nothing ]. As 0.17 is ▼ within the limits below the lower limit above the upper limit of the confidence​ interval, we ▼ cannot can conclude that the population proportion is less than 0.17.

Answer 1

As 0.17 is below the upper limit of the confidence​ interval, so we can conclude that the population proportion is less than 0.17.

Explanation:

We are given that a survey of 270 young professionals found that one dash eighth of them use their cell phones primarily for​ e-mail.

Firstly, the Pivotal quantity for 95% confidence interval for the population proportion is given by;

P.Q. =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion of people who use their cell phones primarily for​ e-mail =
`(1)/(8)` = 0.125

n = sample of young professionals = 270

p = population proportion

Here for constructing 95% confidence interval we have used One-sample z test for proportions.

So, 95% confidence interval for the population proportion, p is ;

P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5% level

of significance are -1.96 & 1.96}

P(-1.96 <
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` < 1.96) = 0.95

P(
`-1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` <
`{\hat p-p}` <
`1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.95

P(
`\hat p-1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` < p <
`\hat p+1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.95

95% confidence interval for p = [
`\hat p-1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ,
`\hat p+1.96 * {\sqrt{(\hat p(1-\hat p))/(n) } }`]

= [
`0.125-1.96 * {\sqrt{(0.125(1-0.125))/(270) } }` ,
`0.125+1.96 * {\sqrt{(0.125(1-0.125))/(270) } }` ]

= [0.08 , 0.16]

Therefore, 95% confidence interval for the population proportion who use cell phones primarily for​ e-mail is [0.08 , 0.16].

Since, the above confidence interval have values which is less than 0.17; so we conclude that the population proportion is less than 0.17.