Ch. 5 - Normal Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.5.28c

Chapter 5, Problem 5.5.28c

Employee Wellness A survey of employed U.S. adults found that only 35% believe their employer cares about their well-being. You randomly select a sample of U.S. employees. Find the probability that fewer than 100 believe their employer cares about their well-being. (Source: Gallup)

c. You select 400 U.S. employees.

Verified step by step guidance

Step 1: Identify the type of probability distribution to use. Since the problem involves a proportion (35%) and a sample size (n = 400), this is a binomial distribution. However, because the sample size is large, we can approximate the binomial distribution using a normal distribution. Verify the conditions for normal approximation: np ≥ 10 and n(1-p) ≥ 10. Here, np = 400 × 0.35 = 140 and n(1-p) = 400 × 0.65 = 260, both of which satisfy the conditions.

Step 2: Calculate the mean (μ) and standard deviation (σ) of the normal distribution. The mean is given by μ = np, and the standard deviation is given by σ = √(np(1-p)). Substitute the values: μ = 140 and σ = √(140 × 0.65).

Step 3: Convert the problem into a z-score calculation. To find the probability that fewer than 100 employees believe their employer cares about their well-being, calculate the z-score using the formula z = (X - μ) / σ, where X = 100, μ = 140, and σ is the standard deviation calculated in Step 2.

Step 4: Use the z-score to find the cumulative probability. Look up the z-score in a standard normal distribution table or use statistical software to find the cumulative probability corresponding to the z-score calculated in Step 3.

Step 5: Interpret the result. The cumulative probability obtained in Step 4 represents the probability that fewer than 100 employees in the sample believe their employer cares about their well-being.

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Key Concepts

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

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. In this context, we can use the binomial distribution since we are dealing with a fixed number of trials (400 employees) and two possible outcomes (believing or not believing that the employer cares). Understanding this distribution is crucial for calculating the probability of fewer than 100 employees expressing concern for their well-being.

Binomial Probability Formula

The binomial probability formula calculates the likelihood of obtaining a specific number of successes in a fixed number of independent Bernoulli trials. It is expressed as P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. This formula will help determine the probability of fewer than 100 employees believing their employer cares.

Normal Approximation to the Binomial

For large sample sizes, the binomial distribution can be approximated by a normal distribution, which simplifies calculations. This approximation is valid when both np and n(1-p) are greater than 5. In this scenario, with 400 employees and a success probability of 0.35, we can use the normal distribution to find the probability of fewer than 100 employees believing their employer cares, making the calculations more manageable.

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Using the Normal Distribution to Approximate Binomial Probabilities

Related Practice

Textbook Question

Approximating Binomial Probabilities In Exercises 19–26, 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. Identify any unusual events. Explain.

Social Media A survey of Americans found that 55% would be disappointed if Facebook disappeared. You randomly select 500 Americans and ask them whether they would be disappointed if Facebook disappeared. Find the probability that the number who say yes is (c) between 240 and 280, inclusive.

Textbook Question

Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.

Health Club Schedule The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (c) more than 30 minutes.

Textbook Question

Daily Commute About 83% of U.S. employees drive their own vehicle to work. You randomly select a sample of U.S. employees. Find the probability that more than 100 of the employees drive their own vehicle to work. (Source: U.S. Bureau of Labor Statistics)

c. You select 150 U.S. employees.

Textbook Question

Uniform Distribution A uniform distribution is a continuous probability distribution for a random variable x between two values a and b (a<b), where (a ≤ x ≤ b) and all of the values of x are equally likely to occur. The graph of a uniform distribution is shown below.

The probability density function of a uniform distribution is

on the interval from (x=a) to (x=b). For any value of x less than a or greater than b, y=0 . In Exercises 59 and 60, use this information.

For two values c and d, where a ≤ c < d ≤ b, the probability that x lies between c and d is equal to the area under the curve between c and d, as shown below.

So, the area of the red region equals the probability that x lies between c and d. For a uniform distribution from (a=1) to (b=25) , find the probability that

d. x lies between 8 and 14.

Textbook Question

Athletes on Social Issues In a survey of college athletes, 84% said they are willing to speak up and be more active in social issues. You randomly select 25 college athletes. Find the probability that the number who are willing to speak up and be more active in social issues is (c) between 18 and 22, inclusive.

Textbook Question

Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.

MCAT Scores In a recent year, the MCAT total scores were normally distributed, with a mean of 500.9 and a standard deviation of 10.6. Find the probability that a randomly selected medical student who took the MCAT has a total score that is (c) more than 515. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)