The Census Bureau's Current Population Survey shows 28% of individuals, ages 25 and older, have completed four years of college (The New York Times Almanac, 2006). For a sample of 15 individuals, ages 25 and older, answer the following questions:
a. What is the probability four will have completed four years of college?
b. What is the probability three or more will have completed four years of college?
Solution
Let $X$ denote the number of individuals ages 25 and older who have completed four years of college. The Census Bureau's Current Population Survey shows 28% of individuals, ages 25 and older, have completed four years of college (The New York Times Almanac, 2006). That is $p = 0.28$ probability that a randomly selected individual with ages 25 and older who have completed four years of college.
Given that $n = 15$ and $p=0.28$. The probability distribution of $X$ is Binomial distribution. That is $X\sim B(15,0.28)$.
The probability mass function of $X$ is
`
$$ \begin{aligned} P(X=x) &= \binom{15}{x} (0.28)^x (1-0.28)^{15-x},\\ & \quad x=0,1,\cdots, 15. \end{aligned} $$
a. The probability that $X$ is exactly $4$ is
$$ \begin{aligned} P(X= 4) & =\binom{15}{4} (0.28)^{4} (1-0.28)^{15-4}\\ & = 0.2262. \end{aligned} $$
The probability four will have completed four years of college $P(X=4) = 0.2262$.
b. the probability three or more will have completed four years of college $3$ is
$$ \begin{aligned} P(X\geq 3) & =1-P(X\leq 2)\\ &= 1-\sum_{x=0}^{2} P(X=x)\\ & = 1-\bigg(0.0072+0.0423 +0.115\bigg) \\ & = 0.8355 \\ \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
