Question

that 45% of adult Americans believe that the overall state of moral values in the United States is poor. (a) Compute the mean and standard deviation of the random variable X, the number of adults who believe that the overall state of moral values in the United States is poor based on a random sample of 500 adult Americans. (b) Interpret the mean. (c) Would it be unusual to identify 240 adult Americans who believe that the overall state of moral values in the United States is poor based on a random sample of 500 adult Americans

Answer 1

Answer:

a) The mean is 225 and the standard deviation is 11.12.

b) It means that in samples of 500 adult Americans, the number of those who believe that the overall state of moral values in the United States is poor is expected to be around 225.

c) No, since 240 is less than 2.5 standard deviations from the mean.

Step-by-step explanation:

For each adult American, there are only two possible outcomes. Either they believe the overall state of moral values in the US is poor, or they do not. The adults are 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.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

In the binomial distribution, a value is considered to be unusual if it is more than 2.5 standard deviations from the mean.

45% of adult Americans believe that the overall state of moral values in the United States is poor.

This means that $p = 0.45$

(a) Compute the mean and standard deviation of the random variable X, the number of adults who believe that the overall state of moral values in the United States is poor based on a random sample of 500 adult Americans.

Sample of 500 means that $n = 500$

Mean:

$E(X) = np = 500*0.45 = 225$

Standard deviation:

$\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{500*0.45*0.55} = 11.12$

The mean is 225 and the standard deviation is 11.12.

(b) Interpret the mean.

It means that in samples of 500 adult Americans, the number of those who believe that the overall state of moral values in the United States is poor is expected to be around 225.

(c) Would it be unusual to identify 240 adult Americans who believe that the overall state of moral values in the United States is poor based on a random sample of 500 adult Americans

2.5 standard deviations above the mean is:

225 + 2.5*11.12 = 252.8

So 253 or more would be unusual, 240 no.