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Ch. 5 - Normal Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 5 - Normal Probability DistributionsProblem 5.R.69b
Chapter 5, Problem 5.R.69b

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.

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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).
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.
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.
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.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


Refer to Exercise 34. A random sample of six days is selected. Find the probability that the mean surface concentration of carbonyl sulfide for the sample is (c) more than 11.1 picomoles per liter. Compare your answers with those in Exercise 34.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


The mean annual salary for Level 1 actuaries in the United States is about \$72,000. A random sample of 45 Level 1 actuaries is selected. What is the probability that the mean annual salary of the sample is (b) more than \(68,000? Assume sigma = \)11,000.

Textbook Question

In Exercises 51 and 52, a population and sample size are given. (b) List all samples (with replacement) of the given size from the population and find the mean of each. (c) Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population.


The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

Textbook Question

In Exercises 61 and 62, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.


A survey of U.S. adults ages 33 to 40 earning more than \$150,000 per year found that 94% are content with how their lives have turned out so far. You randomly select 20 U.S. adults ages 33 to 40 earning more than \$150,000 and ask if they are content with their lives so far.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


The mean annual salary for physical therapists in the United States is about \$87,000. A random sample of 50 physical therapists is selected. What is the probability that the mean annual salary of the sample is (b) more than \(85,000? Assume sigma = \)10,500.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


The mean ACT composite score in a recent year is 20.7. A random sample of 36 ACT composite scores is selected. What is the probability that the mean score for the sample is (a) less than 22, (b) greater than 23, and (c) between 20 and 21.5? Assume sigma=5.9.