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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.Q.8
Chapter 5, Problem 5.Q.8

In a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)
What is the highest score that would still place a person in the bottom 10% of the scores?

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Step 1: Recognize that the problem involves a normal distribution with a mean (μ) of 100 and a standard deviation (σ) of 15. The goal is to find the score (X) that corresponds to the bottom 10% of the distribution.
Step 2: Understand that the bottom 10% corresponds to a cumulative probability (P) of 0.10. This means we are looking for the z-score (z) such that the area to the left of z under the standard normal curve is 0.10.
Step 3: Use a z-table or statistical software to find the z-score corresponding to a cumulative probability of 0.10. From the z-table, the z-score is approximately -1.28.
Step 4: Use the z-score formula to convert the z-score back to the original score (X) in the IQ test distribution. The formula is: X = μ + zσ. Substitute the values: μ = 100, z = -1.28, and σ = 15.
Step 5: Simplify the equation to calculate the value of X. This will give you the highest IQ score that places a person in the bottom 10% of the scores.

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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, indicating that data near the mean are more frequent in occurrence than data far from the mean. In this context, IQ scores follow a normal distribution, which allows us to use statistical methods to determine percentiles and probabilities associated with different scores.
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Percentiles

A percentile is a measure used in statistics indicating the value below which a given percentage of observations fall. For example, the 10th percentile is the score below which 10% of the scores lie. Understanding percentiles is crucial for determining the highest score that still places an individual in the bottom 10% of the distribution.

Z-scores

A Z-score represents the number of standard deviations a data point is from the mean. It is calculated by subtracting the mean from the score and dividing by the standard deviation. In this scenario, Z-scores can be used to find the corresponding IQ score for the 10th percentile, which helps identify the highest score that still falls within that range.
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Related Practice
Textbook Question

In a survey of U.S. adults, 81% feel they have little or no control over data collected about them by companies. You randomly select 250 U.S. adults and ask them whether they feel they have control over data collected about them by companies. Use this information in Exercises 11 and 12. (Source: Pew Research Center)


Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

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Textbook Question

Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (c) less than two. (d) Are any of these events unusual? Explain your reasoning.

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Textbook Question

Find each probability using the standard normal distribution.


b. P(z < 2.23)

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Textbook Question

In a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)


What percent of the IQ scores are greater than 112?

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Textbook Question

Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (b) between 8 and 11, inclusive,

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Textbook Question

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


d. μ = 18.5, σ ≈ 4.25, P(19.6 < x < 26.1)

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