Left-Handed People Ten percent of American adults are left-handed. A statistics class has 25 students in attendance.

a. Find the mean and standard deviation for the number of left-handed students in such classes of 25 students.
b. Would it be unusual to survey a class of 25 students and find that 5 of them are left-handed? Why or why not?

Solution

Let $X$ denote the number of left handed students in a class of 25. Let $p$ be the probability that the selected student is left handed.

Given that $p=0.1$ and $n =25$. Thus $X\sim B(25, 0.1)$.

The probability mass function of $X$ is

$$ \begin{aligned} P(X=x) &= \binom{25}{x} (0.1)^x (1-0.1)^{25-x},\\ & \qquad x=0,1,\cdots, 25\\ & \qquad 0 < p < 1, q=1-p. \end{aligned} $$

(a) The mean for the number of left-handed students in such classes of 25 students is

$$ \begin{aligned} E(X) &= np\\ &= 25\times 0.1 \\ &= 2.5. \end{aligned} $$

The standard deviation for the number of left-handed students in such classes of 25 students is

$$ \begin{aligned} sd &= \sqrt{n*p*(1-p)}\\ &= \sqrt{25\times 0.1\times (1- 0.1)}\\ &= 1.5 \end{aligned} $$

(b) The probability that exactly $5$ is defective is

$$ \begin{aligned} P(X= 5) & =\binom{25}{5} (0.1)^{5} (1-0.1)^{25-5}\\ & = 0.0646\\ \end{aligned} $$

It would not be unusual to survey a class of 25 students and found that 5 of them are left-handed because the probability of $X=5$ is not too small.