A candidate for public office has claimed that 60% of voters will vote for her. If 5 registered voters were sampled,
- What is the probability distribution function for this voting?
- What is the probability that exactly 3 would say they favor this candidate?
- Calculate the mean and standard deviation of the probability distribution.
Solution
Let $X$ denote the number of voters who votes for her out of 5 sampled registered voters. A candidate for public office has claimed that 60% of voters will vote for her. That is $p = 0.60$ probability that a randomly selected candidate vote for her.
- Here $n = 5$ and $p=0.60$. The probability distribution of $X$ is Binomial distribution. That is $X\sim B(5,0.60)$.
The probability mass function of $X$ is
$$ \begin{aligned} P(X=x) &= \binom{5}{x} (0.6)^x (1-0.6)^{5-x},\\ &\qquad \; x=0,1,\cdots, 5. \end{aligned} $$
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- The probability that $X$ is exactly $3$ is
$$ \begin{aligned} P(X= 3) & =\binom{5}{3} (0.6)^{3} (1-0.6)^{5-3}\\ & = 0.3456\\ \end{aligned} $$
The probability that exactly 3 would say they favor this candidate is $P(X=3) = 0.3456$.
- The mean of the probability distribution is $E(X) = np = 5 \times 0.6 = 3$.
The standard deviation of the probability distribution is $sd(X) = \sqrt{np(1-p)} = \sqrt{5 \times 0.6 \times (1- 0.6)} = 1.0954$.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators