A survey found that $64 \%$ of adult Americans operates the flusher of toilets in public restrooms with their foot. Suppose a random sample of $n=500$ adult Americans is obtained and the number who flush public toilets with their foot is recorded.
Find the shape of the binomial probability distribution.
Since $n p(1-p)=$ the shape is
\(\boxed{\text{The shape of the binomial probability distribution is unimodal and skewed to the right.}}\)
Step 1 :Given that the number of trials, denoted as n, is 500 and the probability of success, denoted as p, is 0.64.
Step 2 :The mean of a binomial distribution is given by np. Substituting the given values, we get \(500 \times 0.64 = 320\).
Step 3 :The variance of a binomial distribution is given by np(1-p). Substituting the given values, we get \(500 \times 0.64 \times (1-0.64) = 115.2\).
Step 4 :The mean and variance of the binomial distribution are 320 and 115.2 respectively.
Step 5 :This suggests that the distribution is unimodal (has one peak) and is skewed to the right, since the mean is greater than the median in a binomial distribution.
Step 6 :\(\boxed{\text{The shape of the binomial probability distribution is unimodal and skewed to the right.}}\)