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.


College Acceptance Pennsylvania State University accepts 49% of applicants. You randomly select seven Pennsylvania State University applicants. The random variable represents the number who are accepted. (Source: US News & World Report)

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1. Step 1: Understand the problem and identify the parameters of the binomial distribution. A binomial distribution is defined by two parameters: the number of trials (n) and the probability of success (p). Here, n = 7 (the number of applicants selected) and p = 0.49 (the probability of an applicant being accepted). The random variable x represents the number of applicants accepted.
2. Step 2: Construct the binomial distribution. Use the binomial probability formula to calculate the probability for each possible value of x (from 0 to 7). The formula is: P(x) = (n choose x) * p^x * (1-p)^(n-x), where (n choose x) = n! / [x! * (n-x)!]. Compute P(x) for x = 0, 1, 2, ..., 7.
3. Step 3: Create a histogram to graph the binomial distribution. Plot the values of x (0 through 7) on the horizontal axis and their corresponding probabilities P(x) on the vertical axis. Label the axes clearly and ensure the bars represent the probabilities accurately.
4. Step 4: Describe the shape of the distribution. Analyze the histogram to determine whether the distribution is symmetric, skewed left, or skewed right. For a binomial distribution with p close to 0.5 and a small n, the shape is often approximately symmetric.
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) calculated in Step 2 and identify any x values with probabilities below this threshold. 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 random variable represents the number of applicants accepted from a sample of seven, where each applicant has a 49% chance of being accepted. 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. In the case of a binomial distribution, the histogram will display the frequency of each possible outcome (number of accepted applicants) on the x-axis, with the height of the bars representing the probability of each outcome. This visual tool helps in understanding the shape and spread of the distribution.

Unusual Values

Unusual values in a distribution are those that lie significantly outside the expected range of outcomes, often defined as values that fall beyond two standard deviations from the mean. In the context of the binomial distribution for this problem, identifying unusual values involves analyzing the probabilities of extreme outcomes (like 0 or 7 accepted applicants) and determining if they are significantly lower or higher than what would be expected based on the distribution.