In Exercises 17 and 18, (a) construct a binomial distribution, (b) graph the binomial distribution using a histogram and describe its shape, and (c) identify any values of the random variable x that you would consider unusual. Explain your reasoning.
Seventy-two percent of U.S adults have read a book in any format in the past year. You randomly select five U.S adults and ask them whether they have read a book in any format in the past year. The random variable represents the number of adults who have read a book in any format in the past year. (Source: Pew Research)

Verified step by step guidance

1. Step 1: Understand the problem and identify the parameters of the binomial distribution. The problem involves a binomial experiment where the probability of success (p) is 0.72 (72% of U.S. adults have read a book in the past year), the probability of failure (q) is 1 - p = 0.28, and the number of trials (n) is 5. The random variable x represents the number of adults who have read a book in the past year, and it can take values from 0 to 5.
2. Step 2: Construct the binomial distribution. Use the binomial probability formula: P(x) = (n choose x) * p^x * q^(n-x), where (n choose x) = n! / [x! * (n-x)!]. Calculate the probabilities for each value of x (0, 1, 2, 3, 4, 5) using this formula. For example, for x = 0, P(0) = (5 choose 0) * (0.72)^0 * (0.28)^5. Repeat this for all values of x.
3. Step 3: Create a histogram to graph the binomial distribution. Plot the values of x (0, 1, 2, 3, 4, 5) on the x-axis and their corresponding probabilities P(x) on the y-axis. Use bars to represent the probabilities for each value of x. Observe the shape of the histogram to determine whether it is symmetric, skewed, or follows another pattern.
4. Step 4: Describe the shape of the distribution. Based on the histogram, describe whether the distribution is symmetric, left-skewed, or right-skewed. In this case, since p = 0.72 is greater than 0.5, the distribution is likely to be right-skewed, with higher probabilities concentrated around larger values of x.
5. Step 5: Identify unusual values of x. In statistics, a value is considered unusual if its probability is less than 0.05. Examine the probabilities calculated in Step 2 and identify any values of x for which P(x) < 0.05. Explain why these values are considered unusual based on their low likelihood of occurrence.

Key Concepts

Binomial Distribution

A 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, the success is defined as an adult having read a book in the past year, with a probability of 0.72. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. Each bin's height reflects the frequency of data points within that interval. In the context of the binomial distribution, the histogram will visually depict the probabilities of different outcomes (number of adults who read a book) and help identify the shape of the distribution, which can be symmetric or skewed.

Unusual Values

Unusual values in a statistical context refer to outcomes that lie significantly outside the expected range of results, often defined as those beyond two standard deviations from the mean. In this exercise, identifying unusual values involves analyzing the binomial distribution to determine which counts of adults who have read a book are less likely to occur, thus warranting further investigation or consideration.