Question

a poll of 1132 teens aged 13 to 17 showed that 55% of them have made new friends online. use a 0.05 significance level to test the claim that half of all teens have made new friends online. use the p - value method. use the normal distribution as an approximation to the binomial distribution
let p denote the population proportion of all teens aged 13 to 17 who have made new friends online. identify the null and alternative hypotheses
$h_0: p = 0.5$
$h_1: p
eq 0.5$
(type integers or decimals. do not round.)
identify the test statistic
$z = 3.36$
(round to two decimal places as needed)
identify the p - value
$p - value=square$
(round to three decimal places as needed)

Explanation:

Step1: Determine the type of test

Since \(H_1: p
eq0.5\), this is a two - tailed test.

Step2: Calculate the P - value

For a two - tailed \(z\) - test, the P - value is \(P = 2\times(1 - \Phi(|z|))\), where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution. Given \(z = 3.36\).
We know that \(\Phi(3.36)\approx0.9996\) (from standard normal tables or using a calculator with a normal distribution function).
Then \(P=2\times(1 - 0.9996)\)
\(P = 2\times0.0004=0.0008\approx0.001\)

Answer:

\(0.001\)