Constructing and Graphing Binomial Distributions In Exercises 27–30, (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.


Workplace Cleanliness Fifty-seven percent of employees judge their peers by the cleanliness of their workspaces. You randomly select 10 employees and ask them whether they judge their peers by the cleanliness of their workspaces. The random variable represents the number who judge their peers by the cleanliness of their workspaces. (Source: Adecco)

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 number of trials (n) is 10, the probability of success (p) is 0.57 (since 57% of employees judge their peers by workspace cleanliness), and the random variable (x) represents the number of employees who judge their peers by workspace cleanliness.
2. Step 2: Construct the binomial distribution. Use the binomial probability formula: P(x) = (n choose x) * p^x * (1-p)^(n-x), where (n choose x) = n! / [x! * (n-x)!]. Calculate P(x) for all possible values of x (from 0 to 10). This will give you the probability distribution for the random variable x.
3. Step 3: Create a histogram to graph the binomial distribution. Plot the values of x (0 through 10) on the x-axis and their corresponding probabilities P(x) on the y-axis. Ensure the bars are proportional to the probabilities and label the axes appropriately.
4. Step 4: Describe the shape of the histogram. Based on the probabilities, determine whether the distribution is symmetric, skewed to the left, or skewed to the right. For a binomial distribution with p = 0.57, the histogram is likely to be slightly skewed to the left because p is greater than 0.5.
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 P(x) and identify any x values with probabilities below this threshold. Explain your reasoning based on the calculated probabilities and the context of the problem.

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 random variable represents the number of employees who judge their peers based on workspace cleanliness. The distribution is defined 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 a binomial distribution, a histogram can visually depict the probabilities of different outcomes, helping to illustrate the shape of the distribution.

Unusual Values

Unusual values in a distribution are typically defined as those that lie beyond two standard deviations from the mean. In the context of the binomial distribution, identifying unusual values involves analyzing the probabilities of specific outcomes and determining which values are significantly lower or higher than expected. This helps in understanding the variability and potential outliers in the data.