Textbook Question
In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
To the left of z = 0.72
Ch. 5 - Normal Probability Distributions
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 5, Problem 5.Q.5
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?
Verified step by step guidance1
Identify the key parameters of the normal distribution: the mean (μ = 100) and the standard deviation (σ = 15). The problem asks for the percentage of IQ scores greater than 112.
Standardize the raw score of 112 to a z-score using the formula: , where x is the raw score, μ is the mean, and σ is the standard deviation.
Substitute the values into the z-score formula: . Simplify the numerator and divide by the standard deviation to calculate the z-score.
Use a z-table or a standard normal distribution calculator to find the cumulative probability corresponding to the calculated z-score. This gives the probability of a score being less than 112.
To find the percentage of scores greater than 112, subtract the cumulative probability from 1 (i.e., , where P is the cumulative probability). Multiply the result by 100 to express it as a percentage.
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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. In the context of IQ scores, this means that most individuals score around the average (100), with fewer individuals scoring significantly higher or lower.
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Using the Normal Distribution to Approximate Binomial Probabilities
Z-Score
A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the score and then dividing by the standard deviation. For an IQ score of 112, the Z-score helps determine its position relative to the mean and is essential for finding the percentage of scores above this value.
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06:31Z-Scores From Given Probability - TI-84 (CE) Calculator
Percentile Rank
Percentile rank is a measure used to indicate the value below which a given percentage of observations fall. In this case, calculating the percentile rank for an IQ score of 112 allows us to determine the percentage of individuals who scored lower than this score, which can then be used to find the percentage of scores that are greater.
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02:59Calculating Standard Deviation Example 1
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.
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
The random variable x is normally distributed with the given parameters. Find each probability.
a. μ = 9.2, σ ≈ 1.62, P(x < 5.97)
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)
Textbook Question
The initial pressures for bicycle tires when first filled are normally distributed, with a mean of 70 pounds per square inch (psi) and a standard deviation of 1.2 psi.
b. A random sample of 15 tires is drawn from this population. What is the probability that the mean tire pressure of the sample is less than 69 psi?