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]$n=37 ; p=0.2$[/tex]
Answer (2)
The problem provides a binomial distribution with n = 37 and p = 0.2 .
The formula for the mean of a binomial distribution is μ = n ⋅ p .
Substitute the given values into the formula: μ = 37 ⋅ 0.2 = 7.4 .
The mean of the binomial distribution is 7.4 .
Explanation
Understand the problem and provided data We are given a binomial distribution with parameters n = 37 and p = 0.2 . Our goal is to find the mean, μ , of this binomial distribution and round the answer to the nearest tenth.
Recall the formula for the mean of a binomial distribution The mean of a binomial distribution is given by the formula μ = n ⋅ p , where n is the number of trials and p is the probability of success in each trial.
Substitute the values into the formula Substitute the given values of n and p into the formula: μ = 37 ⋅ 0.2
Calculate the mean Calculate the value of μ : μ = 37 ⋅ 0.2 = 7.4
State the final answer Since the result is already given to one decimal place, no further rounding is needed. Therefore, the mean of the binomial distribution is 7.4 .
Examples
Consider a scenario where you flip a biased coin 37 times, and the probability of getting heads on each flip is 0.2. The mean of the binomial distribution represents the average number of heads you would expect to get over many repetitions of this experiment. In this case, you would expect to get approximately 7.4 heads on average. This concept is useful in various fields such as quality control, where you might inspect a batch of 37 items and want to know the expected number of defective items if the probability of an item being defective is 0.2.
The mean of the binomial distribution with n = 37 and p = 0.2 is calculated using the formula μ = n ⋅ p . Substituting the values gives μ = 7.4 . Therefore, the mean is 7.4 .
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