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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.69c
Chapter 5, Problem 5.R.69c

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 (c) greater than 60.

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Step 1: Verify if the normal distribution can be used to approximate the binomial distribution. For this, check the conditions: (a) The sample size (n) should be large, and (b) 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 = sqrt(np(1-p)). Calculate these values.
Step 3: To find the probability that the number of adults who used a mobile device to manage their bank account is greater than 60, apply the continuity correction. This means you will calculate the probability for X > 60.5 instead of X > 60.
Step 4: Standardize the value using the z-score formula: z = (X - μ) / σ, where X is the value of interest (60.5 in this case), μ is the mean, and σ is the standard deviation. Compute the z-score.
Step 5: Use the standard normal distribution table (or a calculator) to find the probability corresponding to the calculated z-score. Subtract this value from 1 to find the probability that X > 60.5. Sketch the graph of the normal distribution curve, marking the mean and the area representing the probability greater than 60.5.

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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 survey of U.S. adults, where each adult either uses a mobile device to manage their bank account (success) or does not (failure). The parameters of the binomial distribution are the number of trials (n) and the probability of success (p).
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Normal Approximation to the Binomial

The normal approximation to the binomial distribution is used 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 check if the conditions are met to approximate the binomial distribution of mobile device usage with a normal distribution.
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Probability Calculation

Calculating probabilities involves determining the likelihood of a specific outcome occurring within a given distribution. For the binomial distribution, this can be done using the binomial probability formula, while for the normal approximation, we use the z-score to find probabilities from the standard normal distribution. Understanding how to compute these probabilities is essential for answering the question about the number of adults using mobile devices.
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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

The random variable x is normally distributed with the given parameters. Find each probability.

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Textbook Question

In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.


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Textbook Question

In Exercises 51 and 52, a population and sample size are given. (a) Find the mean and standard deviation of the population.

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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 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.