In Exercises 69 and 70, 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. adults found that 72% used a mobile device to manage their bank account at least once in the previous month. You randomly select 70 U.S. adults and ask whether they used a mobile device to manage their bank account at least once in the previous month. Find the probability that the number who have done so is (b) exactly 50.

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) should be large, and (2) both np and n(1-p) should be greater than or equal to 5. Here, n = 70 and p = 0.72. Calculate np = 70 × 0.72 and n(1-p) = 70 × (1 - 0.72).
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: Apply the continuity correction to account for the discrete nature of the binomial distribution. Since we are finding the probability of exactly 50 successes, use the interval [49.5, 50.5] for the normal approximation.
4. Step 4: Standardize the values using the z-score formula: z = (x - μ) / σ, where x represents the boundaries of the interval [49.5, 50.5]. Compute the z-scores for both 49.5 and 50.5.
5. Step 5: Use the standard normal distribution table (or a calculator) to find the probabilities corresponding to the z-scores. Subtract the smaller probability from the larger probability to find the probability of exactly 50 successes. Sketch the graph of the normal curve, shading the area between the z-scores.

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 randomly selecting U.S. adults and determining how many used a mobile device for banking. The parameters include the number of trials (n) and the probability of success (p), which in this case is 0.72.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution is applicable when the number of trials is large, and both np and n(1-p) are greater than 5. This allows us to use the normal distribution to estimate probabilities for binomial outcomes, simplifying calculations. In this case, we would check if the conditions are met to use the normal distribution for approximating the probability of exactly 50 successes.

Probability Calculation

Calculating probability involves determining the likelihood of a specific outcome occurring within a defined set of possibilities. For the binomial distribution, this can be done using the binomial probability formula, which incorporates the number of successes, the total number of trials, and the probability of success. In this exercise, we would calculate the probability of exactly 50 adults using a mobile device for banking using either the binomial formula or the normal approximation if applicable.