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 5.5.23b

Chapter 5, Problem 5.5.23b

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.

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 (b) less than 23

Verified step by step guidance

Step 1: Verify if the normal approximation to the binomial distribution can be used. The conditions are: (1) The sample size n should be large enough, and (2) both np and n(1-p) should be greater than or equal to 5. Here, n = 25 and p = 0.84. Calculate np = 25 * 0.84 and n(1-p) = 25 * (1 - 0.84). Check if both values satisfy the condition.

Step 2: If the conditions are satisfied, approximate the binomial distribution using a 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: Apply the continuity correction for the normal approximation. Since we are finding the probability that the number is less than 23, adjust the value to 22.5 (subtract 0.5 for the continuity correction).

Step 4: Standardize the value using the z-score formula: z = (x - μ) / σ, where x is the adjusted value (22.5), μ is the mean, and σ is the standard deviation. Calculate the z-score.

Step 5: Use the standard normal distribution table (or a calculator) to find the cumulative probability corresponding to the calculated z-score. This will give the probability that the number of athletes willing to speak up is less than 23.

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Key Concepts

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

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p). In this context, the success is defined as a college athlete being willing to speak up on social issues.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution can be used when certain conditions are met, specifically when both np and n(1-p) are greater than 5. This allows for the use of the normal distribution to estimate probabilities for binomial scenarios, simplifying calculations and providing a continuous approximation of the discrete binomial probabilities.

Probability Calculation

Probability calculation involves determining the likelihood of a specific outcome occurring within a defined set of possibilities. In this case, it requires calculating the probability that fewer than 23 out of 25 randomly selected college athletes are willing to engage in social issues, which can be done using either the binomial formula or the normal approximation if applicable.

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Probability From Given Z-Scores - TI-84 (CE) Calculator

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Textbook Question

History Grades In a history class, the grades for various assessments are all positive numbers and have different distributions. Determine whether the grades for each assessment could be normally distributed. Explain your reasoning.

b. a final with a mean of 72, standard deviation of 9, and 90th percentile score of 93

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 (b) between 20 and 28 minutes.

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 (b) between 490 and 510. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)

Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Billboard Hot 100 The length (in seconds) of the 100 most popular songs during the week of May 5, 2021, can be approximated by a normal distribution, as shown in the figure. (Source: Spotify)

b. What song length represents the 17th percentile?

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 (b) at least 300

Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

COVID-19 Response Surveyors asked respondents to rate ten key aspects of their government’s response to the COVID-19 pandemic, including preparedness, communication, and material aid. A pandemic response score that ranged from 0 to 100 was calculated. The mean score for U.S. respondents was 50.6 with a standard deviation of 29.0. (Source: PLOS One)

b. What score represents the 61st percentile?