A recent CBS News survey reported that 67 percent of adults...

A recent CBS News survey reported that 67 percent of adults felt the U.S. Treasury should continue making pennies. Suppose we select a sample of fifteen adults. a. How many of the fifteen would we expect to indicate that the Treasury should continue making pennies? What is the standard deviation? b. What is the likelihood that exactly 8 adults would indicate the Treasury should continue making pennies? c. What is the likelihood at least 8 adults would indicate the Treasury should continue making pennies?

A recent CBS News survey reported that 67 percent of adults felt the U.S. Treasury should continue making pennies.
Suppose we select a sample of fifteen adults.
a. How many of the fifteen would we expect to indicate that the Treasury should continue making pennies? What is the standard deviation?
b. What is the likelihood that exactly 8 adults would indicate the Treasury should continue making pennies?
c. What is the likelihood at least 8 adults would indicate the Treasury should continue making pennies?

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success and failure) and the probability of success remains constant throughout. It is especially useful for modeling survey responses or yes/no experiments, and it underpins the calculations in questions involving expected counts and probabilities for a fixed number of individuals or events.

Expected Value

The expected value in the context of a binomial distribution is calculated as the product of the number of trials and the probability of success on a single trial (E(X) = n * p). It provides a measure of the central tendency or the average outcome one would anticipate over many repetitions of the experiment.

Standard Deviation

The standard deviation in a binomial distribution is a measure of the variability or spread of the number of successes around the expected value. It is computed using the formula ?(n * p * (1 - p)), which captures the dispersion of outcomes based on the number of trials and the success probability.

Binomial Probability Calculation

Binomial probability calculation involves computing the likelihood of achieving exactly a given number of successes in a series of independent trials. This is done using the formula P(X = k) = C(n, k) * p^k * (1 - p)^(n - k), where C(n, k) is the binomial coefficient representing the number of ways to choose k successes out of n trials.

Cumulative Probability

Cumulative probability in the context of a binomial distribution is the probability that the number of successes will fall within a certain range or exceed a given threshold. It is computed by summing the individual binomial probabilities for the desired range of outcomes, for example, calculating the probability of having at least a specific number of successes by summing P(X = k) for all k equal to or greater than that number.

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