If 37% of high school students said that they exercise regularly, find the probability that 5 randomly selected high school students will say that they exercise regularly. Would you consider this event likely or unlikely to occur? Explain your answer.

#### Solution

Let $X$ denote the number of high school students who exercise regularly out of 5 randomly selected students.

Let $p$ denote the probability that high school students exercise regularly.

Given that $p=0.37$.

$X\sim B(n=5, p=0.37)$.

The probability mass function of $X$ is

$$

P(X=x) = \binom{5}{x} (0.37)^x (1-0.37)^{5-x}, \; x=0,1,\cdots, 5.

$$

The probability that 5 randomly selected high school students will say that they exercise regularly is

$$

\begin{aligned}

P(X= 5) & =\binom{5}{5} (0.37)^{5} (1-0.37)^{5-5}\

& = 0.0069\

\end{aligned}

$$

Explanation: The probability 0.0069 is quite low. So it is unlikely to occur that 5 randomly selected high school students will say that they exercise regularly.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators