Problem 57 - Basic Statistics for Business and Economics with Student

Question

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 15 adults. a. How many of the 15 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?

Step-by-Step Solution

Answer

a) Expected: 10.05, Standard deviation: 1.82 b) Likelihood of exactly 8: 0.075 c) Likelihood of at least 8: 0.887

Step 1: Calculate the Expected Number

First, determine how many of the 15 adults are expected to indicate that the Treasury should continue making pennies using the formula for expectation in a binomial distribution, \( E = np \), where \( n = 15 \) and \( p = 0.67 \). So, \( E = 15 \times 0.67 = 10.05 \).

Step 2: Calculate the Standard Deviation

To find the standard deviation in a binomial distribution, use the formula \( \sigma = \sqrt{np(1-p)} \). Substituting \( n = 15 \) and \( p = 0.67 \), we get \( \sigma = \sqrt{15 \times 0.67 \times 0.33} = \sqrt{3.315} \approx 1.82 \).

Step 3: Calculate the Probability of Exactly 8 Adults Responding

To find the probability of exactly 8 adults saying they support penny production, use the binomial probability formula: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n=15 \), \( p=0.67 \), and \( k=8 \).\[ P(X = 8) = \binom{15}{8} (0.67)^8 (0.33)^7 \]Calculate \( \binom{15}{8} = 6435 \), then \( P(X = 8) \approx 6435 \times 0.67^8 \times 0.33^7 \approx 0.075. \)

Step 4: Calculate the Probability of At Least 8 Adults Responding

The probability of at least 8 adults responding means finding \( P(X \geq 8) \). This can be calculated by \( 1 - P(X < 8) \), which is the sum of probabilities from 0 to 7. Calculate each using the binomial formula, similar to Step 3, and sum: \[ P(X \geq 8) = 1 - \sum_{k=0}^{7} \binom{15}{k} (0.67)^k (0.33)^{15-k} \approx 0.887. \]

Key Concepts

Probability Calculations

Probability calculations are essential in determining the likelihood of different outcomes within a set of potential events. In the context of the binomial distribution, we are often interested in finding the probability of a particular number of successes in a series of n independent trials. Here, each trial has two possible outcomes: success (with probability p) or failure (with probability 1-p).

To calculate the probability of exactly k successes in n trials, we use the binomial probability formula:

• \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)

This formula takes into account:

• The number of ways k successes can occur among n trials, expressed as a combination: \( \binom{n}{k} \).
• The probability of success raised to the power of k, \( p^k \).
• The probability of failure raised to the power of the remaining trials, \( (1-p)^{n-k} \).

In our exercise, probabilities are computed by plugging in the values n = 15, p = 0.67, and k = desired number of successes.

This way, probability calculations help us estimate the various outcomes with precision.

Expectation and Variance

In the realm of statistics, the concepts of expectation and variance help us understand the behavior of random variables. Especially within a binomial distribution, they give us meaningful insights about the average outcome and its variability.

The expected value, or expectation, of a binomially distributed random variable X is calculated as \( E = np \). It tells us the average number of successes expected, based on the probability and number of trials. Here, with n = 15 trials and p = 0.67 probability of success, expectation is \( E = 15 \times 0.67 = 10.05 \). This means, on average, about 10 adults are expected to support penny production.

The variance gives us a measure of the spread or dispersion of the distribution. It is calculated as \( \text{Var}(X) = np(1-p) \). This shows how much variability we have around the expected value. For our exercise, substituting n = 15 and p = 0.67, variance equals \( 3.315 \).

The standard deviation, the square root of variance, offers a more intuitive measure of spread, approximated here as \( \sigma \approx 1.82 \). It indicates how much outcomes are expected to deviate from the mean, either positively or negatively, in typical scenarios.

Probability Mass Function

The probability mass function (PMF) is a fundamental concept when dealing with discrete probability distributions like the binomial distribution. It provides the probability that a discrete random variable is exactly equal to a particular value.

In a binomial context, the PMF helps compute probabilities of achieving a specific number of successes in a series of trials. Using the binomial formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), we can derive the PMF for certain outcomes. For example, knowing the PMF allows us to calculate the probability of exactly 8 adults supporting pennies: \( P(X = 8) \approx 0.075 \) in our scenario.

This function is key to solving various probability problems because it specifies probabilities for each potential outcome, ranging from 0 to the total number of trials, n. By summing these discrete probabilities, it is also possible to find cumulative probabilities. For instance, the likelihood of at least 8 adults supporting penny production can be evaluated by summing probabilities from 8 to 15.

• This teaches us how to leverage the PMF for precise probability queries.