Finding Binomial Probabilities In Exercises 19–26, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B.

Responsible Consumption Forty-five percent of consumers say it is important that the clothing they buy is made without child labor. You randomly select 16 consumers. Find the probability that the number of of consumers who say it is important that the clothing they buy is made without child labor is (a) e

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Step 1: Identify the problem type. This is a binomial probability problem because we are dealing with a fixed number of trials (16 consumers), two possible outcomes (say it is important or not), and a constant probability of success (45% or 0.45).

Step 2: Define the binomial random variable. Let X represent the number of consumers who say it is important that the clothing they buy is made without child labor. X follows a binomial distribution with parameters n = 16 (number of trials) and p = 0.45 (probability of success).

Step 3: Write the formula for the binomial probability. The probability mass function for a binomial random variable is given by:

P (X = k) = (\frac{n!}{k! (n - k)!}) \times p^{k} \times (^{1 - p) n - k}

, where k is the number of successes, n is the number of trials, and p is the probability of success.

Step 4: Plug in the values for n and p into the formula. For this problem, n = 16 and p = 0.45. To find the probability for a specific value of k (the number of successes), substitute k into the formula and calculate the result.

Step 5: Use technology or a binomial probability table to compute the probability for the desired value of k. If using technology, such as a calculator or statistical software, input the values for n, p, and k to obtain the probability. Alternatively, refer to Table 2 in Appendix B for the binomial probabilities.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, it applies to the scenario of selecting consumers who value child labor-free clothing, where each consumer's response can be seen as a success or failure.

Probability Mass Function (PMF)

The probability mass function gives the probability of obtaining exactly k successes in n trials for a binomial distribution. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success. This function is essential for determining the likelihood of a specific number of consumers expressing the importance of child labor-free clothing.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain threshold. In the context of this question, it may be necessary to calculate cumulative probabilities to find the likelihood of a certain number of consumers valuing child labor-free clothing, which can be done using binomial tables or technology.

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