In Exercises 5 and 6, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.


A survey of U.S. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (b) less than 5. Identify any unusual events. Explain.

Verified step by step guidance

1. Step 1: Verify if the normal distribution can be used to approximate the binomial distribution. For this, check the conditions: (1) The sample size (n) must be large enough, and (2) both np and n(1-p) must be greater than or equal to 5. Here, n = 18 and p = 0.37. Calculate np = 18 * 0.37 and n(1-p) = 18 * (1 - 0.37).
2. Step 2: If the conditions are satisfied, proceed to approximate the binomial distribution using the normal distribution. The mean (μ) and standard deviation (σ) of the binomial distribution are given by μ = np and σ = √(np(1-p)). Calculate these values using the given n and p.
3. Step 3: To find the probability that the number of students who prefer to take a job in a different state is less than 5, convert the binomial random variable to a z-score using the formula z = (x - μ) / σ, where x is the value of interest. Since the problem asks for 'less than 5,' use x = 4.5 (apply the continuity correction).
4. Step 4: Use the z-score obtained in Step 3 to find the corresponding probability from the standard normal distribution table or a statistical software. This will give the approximate probability for the event.
5. Step 5: Identify any unusual events by comparing the calculated probability to a threshold (e.g., 0.05 for a 5% significance level). If the probability is less than the threshold, the event is considered unusual. Additionally, sketch the graph of the normal distribution curve with the area corresponding to the probability shaded.

Key Concepts

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 undergraduates who prefer to take a job in a different state, where each student represents a trial with a success probability of 0.37.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution is applicable when the number of trials is large, and both the expected number of successes and failures are greater than 5. This allows us to use the normal distribution to estimate probabilities for binomial scenarios, simplifying calculations and providing a continuous approximation.

Unusual Events

An unusual event in statistics is typically defined as one that has a low probability of occurring, often less than 5%. In this problem, identifying whether the event of having fewer than 5 students preferring a job out of state is unusual involves calculating its probability using either the binomial or normal distribution and comparing it to this threshold.