Question

Suppose that a recent poll found that 57% of adults believe that the overall state of moral values is poor. Complete parts (a) through (c). 

(a) For 400 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 The standard deviation of X is 

(b) Interpret the mean. Choose the correct answer below. 

A. For every 400 adults, the mean is the minimum number of them that would be expected to believe that the overall state of moral values is poor. 

B. For every 400 adults, the mean is the number of them that would be expected to believe that the overall state of moral values is poor. 

C. For every 400 adults, the mean is the range that would be expected to believe that the overall state of moral values is poor. 

D. For every 228 adults, the mean is the maximum number of them that would be expected to believe that the overall state of moral values is poor. 

(c) Would it be unusual if 215 of the 400 adults surveyed believe that the overall state of moral values is poor?  No  Yes

Answer 1

Concepts and reason

A binomial experiment is a discrete probability experiment that satisfies the following conditions:

The experiment is repeated for a fixed number of trails, each trail is independent of the other trails.

There are two possible outcomes for each trail. The outcomes are classified as success (S) or as a failure (F).

The probability of success, p remains same for each trail. The random variable X represents the number of successes in n independent trails of the experiment.

Fundamentals

The probability of success is known and number of trails is fixed. So apply binomial distribution.

Let the random variable X follows a binomial distribution with parameters n and p.

The probability mass function (PMF) of binomial distribution can be defined as,

The mean and standard deviation of the random variable X is,

(a)

From the given information, moral values rich or poor are independent of one another.

The possible outcomes are “rich” or “poor.”

Let the random variable X represents the number of adults who believe that the overall state of moral values is poor.

Let n is the number of randomly selected adults participated in the recent poll.

Let p=0.57 represent the proportion of adults believe that the overall state of moral values is poor.

The experiment is repeated n times (countable finite).

The mean of the random variable X is,

The standard deviation of the random variable X is,

(b)

The objective is to interpret mean of the random variable X.

The expected value of the random variable is also known as the mean of the random variable.

Hence, the mean is the number of them that would be expected to believe that the overall state of moral values is poor for every 400 adults.

(c)

From the given information,

Let x=215 represent the number of adults surveyed believes that the overall state of moral values is poor.

Let n=400 represent the number of randomly selected adults participated in the recent poll.

Hence, the proportion of adults surveyed believes that the overall state of moral values is poor is,

Here, the probability is greater than 0.05.

Therefore, it would not be unusual.

Ans: Part a

The mean and standard deviation of the random variable X is 228 and 9.90.