Find the mean, [tex]$\mu$[/tex], for the binomial distribution which has the stated values of n and p. Round answer to the nearest tenth.
[tex]$n=32 ; p=3 / 5$[/tex]
Answer (1)
The problem provides a binomial distribution with n = 32 and p = 5 3 .
The formula for the mean of a binomial distribution is μ = n ⋅ p .
Substitute the values into the formula: μ = 32 ⋅ 5 3 = 19.2 .
The mean of the binomial distribution, rounded to the nearest tenth, is 19.2 .
Explanation
Understand the problem and provided data We are given a binomial distribution with parameters n = 32 and p = 5 3 . We need to find the mean, μ , of this binomial distribution and round the answer to the nearest tenth.
State the formula for the mean The mean of a binomial distribution is given by the formula μ = n ⋅ p .
Substitute the values Substitute the given values of n and p into the formula: μ = 32 ⋅ 5 3 .
Calculate the mean Calculate the value of μ : μ = 32 ⋅ 5 3 = 5 96 = 19.2 .
State the final answer Since the result is already given to the nearest tenth, no further rounding is needed. Therefore, the mean of the binomial distribution is 19.2 .
Examples
Consider a quality control process where you inspect 32 items from a production line, and the probability that any one item is defective is 3/5. The mean of the binomial distribution tells you the average number of defective items you would expect to find in the sample. This helps in assessing the overall quality and making informed decisions about process improvements. For example, if the mean number of defective items is unacceptably high, you might need to adjust the production process to reduce the defect rate. In this case, you would expect to find approximately 19.2 defective items on average.
