Question

A Pew Internet and American Life Project survey found that 392 of 799 randomly selected teens reported texting with their friends every day. Calculate and interpret a 95% confidence interval for the population proportion p that would report texting with their friends every day.

Answer 1

To calculate a 95% confidence interval for the population proportion p in this case, we can use the formula: CI = p ± Z * sqrt((p * (1-p))/n). Plugging in the values, the confidence interval is: 0.4622 to 0.5189.

Step-by-step explanation:

To calculate a confidence interval for the population proportion p, we can use the formula:

CI = p ± Z * sqrt((p * (1-p))/n)

In this case, the sample proportion is 392/799 = 0.4906. We want a 95% confidence interval, so the Z-value is 1.96 (from a standard normal distribution). The sample size, n, is 799. Plugging these values into the formula, we get:

CI = 0.4906 ± 1.96 * sqrt((0.4906 * (1-0.4906))/799)

Calculating the values, we get:

CI = 0.4906 ± 0.0284

This means we can be 95% confident that the true proportion of teens who report texting with their friends every day is between 0.4622 and 0.5189.

Answer 2

Answer with explanation:

The confidence interval for population proportion (p) is given by :-

`\hat{p}\pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}`

, where
`\hat{p}` = sample proportion

n= Sample size

z*= Critical z-value.

Let p = population proportion of teens that would report texting with their friends every day.

As per given , we have

n=799

`\hat{p}=(392)/(799)=0.49`

Critical z value for 95% confidence = z* =1.96

Now , the 95% confidence interval for the population proportion p that would report texting with their friends every day. would be:

`0.49\pm 1.96\sqrt{(0.49(1-0.49))/(799)}`

`0.49\pm 1.96√(0.000312765957447)`

`0.49\pm 1.96(0.01768519)`

`0.49\pm 0.03466 =(0.49-0.03466,\ 0.49+0.03466)`

`=(0.45534,\ 0.52466)\approx(0.46,\ 0.52)`

95% confidence interval for the population proportion p: (0.46, 0.52)

Interpretation: We are 95% confident that the true population proportion p of teens that would report texting with their friends every day lies in (0.46, 0.52).