/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 The Los Angeles Times (July 17,1... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 67

The Los Angeles Times (July 17,1995 ) reported that in a sample of 364 lawsuits in which punitive damages were awarded, the sample median damage award was \(\$ 50,000\), and the sample mean was \(\$ 775,000\). What does this suggest about the distribution of values in the sample?

Short Answer

Expert verified
The distribution of values in the sample is positively skewed. This is due to the very high punitive damage awards pulling up the mean, indicating that there are some exceptionally large damage award values in the sample.

Step by step solution

01

Understanding the Dataset

The dataset we are dealing with is a sample of 364 lawsuits in which punitive damages were awarded. The value in this case is the amount of the punitive damages. The sample median damage award is \$50,000 and the sample mean is \$775,000.
02

Distinguishing Median and Mean

In a sorted dataset, the median is the middle number. When the numbers are uneven, the median is the direct middle number. If even, it鈥檚 the average of the two middle numbers. In this case, the median damage award is \$50,000. The mean is the sum of the values divided by the number of values, which here comes out to \$775,000.
03

Drawing Implications

The mean is highly influenced by outliers (exceptionally high or low values) in the dataset, and that can increase or decrease it. In this case, the mean is much higher than the median. This suggests that there are very high punitive damage awards in the sample pull up the mean. Therefore, the distribution of values in the sample is skewed to the right or positively skewed.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Median
The median is a measure of central tendency that helps us understand the middle value of a dataset. It's especially useful when you want to know what the typical value is, without being affected by extremely high or low numbers that might skew the result. In our case, with the punitive damages, the median value is given as \(\\(50,000\).
This means that half of the lawsuits awarded damages less than \(\\)50,000\) and the other half awarded more. The median doesn't tell us about the extreme values, but rather gives us a clear picture of the center point of the dataset.
Unlike the mean, the median isn't influenced by outliers, making it a reliable indicator for skewed data.
Mean
The mean, or average, is another essential measure of central tendency. You find it by adding up all the numbers in a dataset and then dividing by the count of the numbers. In this example of lawsuits, the mean punitive damage award is \(\$775,000\).
The mean takes every value into account, which means it can be affected by unusually high or low values known as outliers. If a few of the lawsuits awarded exceptionally high damages, this will bump up the mean compared to the median, indicating that the dataset might be skewed.
Understanding the mean can provide insight into the overall dataset, but it's important to consider its sensitivity to outliers.
Skewness
Skewness refers to the asymmetry of a distribution of values. It tells us about the direction and degree of asymmetry in a dataset compared to a normal distribution. In our example with the punitive damages, the mean is much higher than the median. This suggests that there are some very high values in the dataset.
When the mean is significantly greater than the median, it indicates a positive skew or right skewness. This means a bulk of the values lie on the lower end, and there are some extreme higher values pulling the average up.
  • Positive Skew: Mean > Median.
  • Negative Skew: Mean < Median.
  • No Skew: Mean 鈮� Median.
Understanding skewness helps to visualize how data is spread and to anticipate potential outliers or extreme values.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that the distribution of scores on an exam is closely described by a normal curve with mean 100 . The 1 6th percentile of this distribution is 80 . a. What is the 84 th percentile? b. What is the approximate value of the standard deviation of exam scores? c. What \(z\) score is associated with an exam score of 90 ? d. What percentile corresponds to an exam score of 140 ? e. Do you think there were many scores below 40 ? Explain.

O Although bats are not known for their eyesight, they are able to locate prey (mainly insects) by emitting high-pitched sounds and listening for echoes. A paper appearing in Animal Behaviour ("The Echolocation of Flying Insects by Bats" [1960]: \(141-154\) ) gave the following distances (in centimeters) at which a bat first detected a nearby insect: $$ \begin{array}{lllllllllll} 62 & 23 & 27 & 56 & 52 & 34 & 42 & 40 & 68 & 45 & 83 \end{array} $$ a. Compute the sample mean distance at which the bat first detects an insect. b. Compute the sample variance and standard deviation for this data set. Interpret these values.

The paper cited in Exercise \(4.61\) also reported values of single-leg power for a low workload. The sample mean for \(n=13\) observations was \(\bar{x}=119.8\) (actually 119.7692), and the 14 th observation, somewhat of an outlier, was 159 . What is the value of \(\bar{x}\) for the entire sample?

The standard deviation alone does not measure relative variation. For example, a standard deviation of \(\$ 1\) would be considered large if it is describing the variability from store to store in the price of an ice cube tray. On the other hand, a standard deviation of \(\$ 1\) would be considered small if it is describing store-to-store variability in the price of a particular brand of freezer. A quantity designed to give a relative measure of variahility is the \(\mathrm{co-}\) efficient of variation. Denoted by CV, the coefficient of variation expresses the standard deviation as a percentage of the mean. It is defined by the formula \(C V=100\left(\frac{s}{\bar{x}}\right)\). Consider two samples. Sample 1 gives the actual weight (in ounces) of the contents of cans of pet food labeled as having a net weight of 8 oz. Sample 2 gives the actual weight (in pounds) of the contents of bags of dry pet food labeled as having a net weight of \(50 \mathrm{lb}\). The weights for the two samples are: \(\begin{array}{lrrrrr}\text { Sample 1 } & 8.3 & 7.1 & 7.6 & 8.1 & 7.6 \\ & 8.3 & 8.2 & 7.7 & 7.7 & 7.5 \\ \text { Sample 2 } & 52.3 & 50.6 & 52.1 & 48.4 & 48.8 \\ & 47.0 & 50.4 & 50.3 & 48.7 & 48.2\end{array}\) a. For each of the given samples, calculate the mean and the standard deviation. b. Compute the coefficient of variation for each sample. Do the results surprise you? Why or why not?

An advertisement for the "30-in. Wonder" that appeared in the September 1983 issue of the journal Packaging claimed that the 30 -in. Wonder weighs cases and bags up to \(110 \mathrm{lb}\) and provides accuracy to within \(0.25 \mathrm{oz}\). Suppose that a 50 -oz weight was repeatedly weighed on this scale and the weight readings recorded. The mean value was \(49.5 \mathrm{oz}\), and the standard deviation was \(0.1\). What can be said about the proportion of the time that the scale actually showed a weight that was within \(0.25 \mathrm{oz}\) of the true value of 50 oz? (Hint: Use Chebyshev's Rule.)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.