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 5.3.32b

Chapter 5, Problem 5.3.32b

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?

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Step 1: Understand the problem. You are tasked with finding the score that corresponds to the 61st percentile in a normal distribution. The mean (μ) is 50.6, and the standard deviation (σ) is 29.0. Percentiles represent the cumulative probability in a normal distribution.

Step 2: Convert the percentile (61%) into a z-score using the standard normal distribution table or a statistical tool. The z-score is a standardized value that corresponds to the cumulative probability of 0.61 in the standard normal distribution.

Step 3: Use the z-score formula to find the raw score (X) in the original distribution. The formula is:

X = μ + z \times σ

, where μ is the mean, σ is the standard deviation, and z is the z-score.

Step 4: Substitute the values into the formula. Use μ = 50.6, σ = 29.0, and the z-score obtained from Step 2.

Step 5: Calculate the raw score (X) using the formula. This will give you the score that represents the 61st percentile in the given normal distribution.

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Key Concepts

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

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. Understanding this concept is crucial for interpreting data in many fields, including statistics, as it helps in making inferences about population parameters based on sample statistics.

Percentiles

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, the 61st percentile is the score below which 61% of the data points lie. This concept is essential for understanding the relative standing of a score within a distribution, allowing for comparisons between different scores and the overall distribution of data.

Z-scores

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It indicates how many standard deviations an element is from the mean. Z-scores are particularly useful in normal distributions for determining percentiles, as they allow for the conversion of raw scores into a standardized form that can be easily compared across different datasets.

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Related Practice

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

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.

Advancing Research In a survey of U.S. adults, 77% said are willing to share their personal health information to advance medical research. You randomly select 500 U.S. adults. Find the probability that the number who are willing to share their personal health information to advance medical research is (c) between 380 and 390 inclusive.

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 (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.

Weights of Teenagers In a survey of 18-year old males, the mean weight was 166.7 pounds with a standard deviation of 49.3 pounds. (Adapted from National Center for Health Statistics)

c. What weight represents the first quartile?