Textbook Question
Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.
Ch. 6 - Normal Probability Distributions
Triola14th EditionElementary StatisticsISBN: 9780137366446
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Table of contents
- Ch. 1 - Introduction to Statistics88
- Ch. 2 - Exploring Data with Tables and Graphs70
- Ch. 3 - Describing, Exploring, and Comparing Data66
- Ch. 4 - Probability112
- Ch. 5 - Discrete Probability Distributions109
- Ch. 6 - Normal Probability Distributions122
- Ch. 7 - Estimating Parameters and Determining Sample Sizes107
- Ch. 8 - Hypothesis Testing75
- Ch. 9 - Inferences from Two Samples94
- Ch. 10 - Correlation and Regression80
- Ch. 11 - Goodness-of-Fit and Contingency Tables29
- Ch. 12 - Analysis of Variance42
Chapter 6, Problem 6.2.38
Outliers For the purposes of constructing modified boxplots as described in Section 3-3, outliers are defined as data values that are above Q3 by an amount greater than 1.5 x IQR or below Q1 by an amount greater than 1.5 x IQR, where IQR is the interquartile range. Using this definition of outliers, find the probability that when a value is randomly selected from a normal distribution, it is an outlier.
Verified step by step guidance1
Step 1: Recall the definition of an outlier. An outlier is a data value that is either greater than Q3 + 1.5 × IQR or less than Q1 - 1.5 × IQR, where Q1 and Q3 are the first and third quartiles, respectively, and IQR (interquartile range) is defined as Q3 - Q1.
Step 2: For a normal distribution, the quartiles Q1 and Q3 correspond to specific z-scores. Q1 corresponds to the 25th percentile, and Q3 corresponds to the 75th percentile. Using a z-score table or standard normal distribution properties, these z-scores are approximately -0.674 and 0.674, respectively.
Step 3: Calculate the IQR in terms of z-scores. Since IQR = Q3 - Q1, substitute the z-scores: IQR = 0.674 - (-0.674) = 1.348.
Step 4: Determine the z-score boundaries for outliers. The boundaries are Q3 + 1.5 × IQR and Q1 - 1.5 × IQR. Substituting the values: Upper boundary = 0.674 + 1.5 × 1.348, and Lower boundary = -0.674 - 1.5 × 1.348.
Step 5: Use the standard normal distribution to find the probability of being outside these boundaries. This involves calculating the cumulative probability for the lower boundary and subtracting it from 1 minus the cumulative probability for the upper boundary. Use a z-score table or statistical software to find these probabilities.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Outliers
Outliers are data points that significantly differ from other observations in a dataset. In the context of boxplots, they are defined as values that lie beyond the thresholds set by the first quartile (Q1) and the third quartile (Q3), specifically those that are more than 1.5 times the interquartile range (IQR) above Q3 or below Q1. Identifying outliers is crucial for understanding data distribution and ensuring accurate statistical analysis.
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Interquartile Range (IQR)
The interquartile range (IQR) is a measure of statistical dispersion that represents the range within which the central 50% of data points lie. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). The IQR is particularly useful for identifying outliers, as it provides a robust measure of variability that is less affected by extreme values compared to the overall range.
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Normal Distribution
A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, where most observations cluster around the mean, and probabilities for values further away from the mean taper off symmetrically. In a normal distribution, approximately 68% of values fall within one standard deviation of the mean, and about 95% fall within two standard deviations. Understanding the properties of normal distribution is essential for calculating probabilities related to outliers.
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Related Practice
Textbook Question
Car Colors
In Exercises 9–12, assume that 100 cars are randomly selected. Refer to the accompanying graph, which shows the top car colors and the percentages of cars with those colors (based on PPG Industries).
Black Cars Find the probability that at least 25 cars are black. Is 25 a significantly high number of black cars?
Textbook Question
Determining Normality. In Exercises 9–12, refer to the indicated sample data and determine whether they appear to be from a population with a normal distribution. Assume that this requirement is loose in the sense that the population distribution need not be exactly normal, but it must be a distribution that is roughly bell-shaped.
Taxi Trips The distances (miles) traveled by New York City taxis transporting customers, as listed in Data Set 32 “Taxis” in Appendix B
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Textbook Question
Basis for the Range Rule of Thumb and the Empirical Rule. In Exercises 45–48, find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About __ % of the area is between z = -3.5 and z = 3.5 (or within 3.5 standard deviation of the mean).
Textbook Question
Constructing Normal Quantile Plots. In Exercises 17–20, use the given data values to identify the corresponding z scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile plot, then determine whether the data appear to be from a population with a normal distribution.
Earthquake Depths A sample of depths (km) of earthquakes is obtained from Data Set 24 “Earthquakes” in Appendix B: 17.3, 7.0, 7.0, 7.0, 8.1, 6.8.
Textbook Question
Finding Bone Density Scores. In Exercises 37–40 assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.
Find P99, the 99th percentile. This is the bone density score separating the bottom 99% from the top 1%.