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 5.3.31c

Chapter 5, Problem 5.3.31c

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?

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Identify the key parameters of the normal distribution: the mean (μ) is 166.7 pounds, and the standard deviation (σ) is 49.3 pounds. The first quartile corresponds to the 25th percentile of the distribution.

Recall that for a normal distribution, the z-score corresponding to the 25th percentile can be found using a z-table or statistical software. The z-score for the 25th percentile is approximately -0.674.

Use the z-score formula to find the weight (x) corresponding to the first quartile: z = (x - μ) / σ. Rearrange the formula to solve for x: x = μ + z * σ.

Substitute the known values into the formula: x = 166.7 + (-0.674) * 49.3. This will give the weight corresponding to the first quartile.

Perform the calculation to find the weight. This value represents the first quartile of the distribution, meaning 25% of the weights are below this value.

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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, showing 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 analyzing data that follows this distribution, as it allows for the application of various statistical methods.

Quartiles

Quartiles are values that divide a dataset into four equal parts, each containing 25% of the data points. The first quartile (Q1) is the median of the lower half of the dataset, representing the 25th percentile. Knowing how to calculate quartiles is essential for understanding the distribution of data and identifying outliers or trends within a dataset.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In the context of normal distribution, it helps in determining the spread of data points and is vital for calculating quartiles and other statistical measures.

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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.

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Textbook Question

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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)

137

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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?

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