Question

Suppose that a recent poll found that 49​% of adults believe that the overall state of moral values is poor. Complete parts​ (a) through​ (c). ​(a) For 250 randomly selected​ adults, compute the mean and standard deviation of the random variable​ X, the number of adults who believe that the overall state of moral values is poor. The mean of X is nothing.​ (Round to the nearest whole number as​ needed.) The standard deviation of X is nothing. ​(Round to the nearest tenth as​ needed.)

Answer 1

The mean of X is 122.5 and the standard deviation is 7.9.

Explanation:

For each adult, there are only two possible outcomes. Either they believe that the overall state of moral values is poor, or they do not believe this. The probability of an adult believing this is independent of other adults. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

In this problem, we have that:

`n = 250, p = 0.49`

So

`E(X) = np = 250*0.49 = 122.5`

`√(V(X)) = √(np(1-p)) = √(250*0.49*0.51) = 7.9`

The mean of X is 122.5 and the standard deviation is 7.9.