Question

Researchers are concerned about the rise in drinking behaviors among adolescents aged 13-17 in a particular community. Suppose the research team estimates the prevalence of such behaviors is 17%. A sample of 12 adolescents in this age range is collected. Let X represent the number of individuals in the sample with presence of this behavior. • The probability of seeing at least 2 cases in a sample 0.7360 0.2627 0.6304 0.3696 None of the other choices represents a suitable response

Answer 1

X be the number of individual in the sample with presence of this behaviour

X ~ Binomial with p = 17% = 0.17

and n = 12

We need to find

Therefore, we get that $\Pr(X \ge 2)$ = 1 - $\Pr(X \le 1)$

$\Pr(X \le 1) = \sum_{i=0}^{ 1} \Pr(X = i) = \sum_{i=0}^{ 1} \left( \begin{matrix} n \\ i \end{matrix}\right) p^i (1-p)^{n-i}$

This implies that

$\Pr(X \le 1) = \Pr(X = 0) + \Pr(X = 1)$

$\Pr(X \le 1)= \left( \begin{matrix} 12 \\ 0 \end{matrix}\right) 0.17^{ 0} \times 0.83^{ 12-0} + \left( \begin{matrix} 12 \\ 1 \end{matrix}\right) 0.17^{ 1} \times 0.83^{ 12-1}$

$\Pr(X \le 1) =0.1069+0.2627$

$\Pr(X \le 1)= 0.3696$

$\Pr(X \ge 2)$ = 1 - 0.3696

$\Pr(X \ge 2)$ = 0.6304