/*! 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 51 Arsenic is toxic to humans, and ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 51

Arsenic is toxic to humans, and people can be exposed to it through contaminated drinking water, food, dust, and soil. Scientists have devised an interesting new way to measure a person's level of arsenic poisoning: by examining toenail clippings. In a recent study, \({ }^{29}\) scientists measured the level of arsenic (in \(\mathrm{mg} / \mathrm{kg}\) ) in toenail clippings of eight people who lived near a former arsenic mine in Great Britain. The following levels were recorded: \(\begin{array}{ll}0.8 & 1.9\end{array}\) \(\begin{array}{llll}3.9 & 7.1 & 11.9 & 26.0\end{array}\) \(\begin{array}{ll}2.7 & 3.4\end{array}\) (a) Do you expect the mean or the median of these toenail arsenic levels to be larger? Why? (b) Calculate the mean and the median.

Short Answer

Expert verified
The mean arsenic level is 7.34 mg/kg and the median arsenic level is 3.65 mg/kg. You would expect the mean to be larger because it is more sensitive to outliers or extremely high values in the data.

Step by step solution

01

Understand what mean and median are

The mean is the sum of the sample divided by the number of elements in the sample. The median is the middle value of a list when it is ordered in either increasing or decreasing order. If there's an even number of observations, the median will be the average of the two middle numbers.
02

Order the data and calculate the median

First, arrange the data in ascending order: 0.8, 1.9, 2.7, 3.4, 3.9, 7.1, 11.9, 26.0. Since we have 8 observations, the median is the average of the fourth and fifth observations. Therefore, the median is (3.4+3.9)/2 = 3.65 mg/kg.
03

Calculate the mean

Calculate the mean by summing all the data points and dividing by the number of data points: (0.8 + 1.9 + 2.7 + 3.4 + 3.9 + 7.1 + 11.9 + 26.0) / 8 = 7.34 mg/kg.
04

Compare the mean and the median

Comparing these results, it is clear that the mean (7.34 mg/kg) is larger than the median (3.65 mg/kg). A one-off high value of 26.0 mg/kg drives the mean up. However, because the median only considers the middle data points in an ordered list, it is not so affected by potentially skewed data points, making it a more balanced representation of the given data in this case.

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.

Mean vs Median
When we begin analyzing data, it's crucial to understand two key measures of central tendency: the mean and the median. The mean, often referred to as the average, is computed by summing up all the values in a dataset and then dividing that sum by the number of data points. It represents the central point of a data set.

The median, on the other hand, is the middle value when a data set is ordered from smallest to largest (or vice versa). If there's an even number of observations, the median is the midway point between the two central numbers, found by computing their average.

In the context of the arsenic levels in toenail clippings, we would expect the mean to be higher if there are outliers or extremely high values, as these would skew the mean upwards. The median, unaffected by such anomalies, would likely provide a more representative value of the central tendency in this scenario.
Data Analysis
Data analysis encompasses a variety of techniques to inspect, clean, transform, and model data with the goal of discovering useful information, informing conclusions, and supporting decision-making. One of the primary steps in data analysis is understanding the distribution of data points.

For the arsenic exposure study, by calculating both the mean and median, scientists can gain insights about the data distribution. If the mean is significantly higher than the median, as is the case with the arsenic levels, it suggests that the data is right-skewed and that there are a few very high values affecting the mean.
Measures of Central Tendency
Measures of central tendency are statistical tools used to summarize a set of data by identifying the center point of its distribution. The three main measures are the mean, median, and mode. The mean provides a mathematical average, the median presents the middle value, and the mode is the most frequently occurring value in a data set.

The choice between these measures depends on the nature of the data and the specific information sought. For data with outliers or a skewed distribution, the median can often be a more reliable measure than the mean, as it doesn't get distorted by extreme values.
Outliers in Data
Outliers are data points that differ significantly from other observations. They can occur due to variability in the measurement or possibly due to experimental error. Outliers can have a pronounced effect on the mean, pulling it toward their value and potentially resulting in a misleading interpretation of the data.

In our exercise, the high arsenic level of 26.0 mg/kg acts as an outlier, causing the mean to misrepresent the typical arsenic exposure. Identifying outliers is an essential part of data analysis, as it can influence which measure of central tendency (mean, median, or mode) best represents the typical data point within the dataset.

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

For each set of data in Exercises 2.43 to 2.46: (a) Find the mean \(\bar{x}\). (b) Find the median \(m\). (c) Indicate whether there appear to be any outliers. If so, what are they? \(\begin{array}{llllllll}\mathbf15, & 22, & 12, & 28, & 58, & 18, & 25, & 18\end{array}\)

Use the \(95 \%\) rule and the fact that the summary statistics come from a distribution that is symmetric and bell-shaped to find an interval that is expected to contain about \(95 \%\) of the data values. A bell-shaped distribution with mean 10 and standard deviation 3

The Impact of Strong Economic Growth In 2011, the Congressional Budget Office predicted that the US economy would grow by \(2.8 \%\) per year on average over the decade from 2011 to 2021 . At this rate, in \(2021,\) the ratio of national debt to GDP (gross domestic product) is predicted to be \(76 \%\) and the federal deficit is predicted to be \(\$ 861\) billion. Both predictions depend heavily on the growth rate. If the growth rate is \(3.3 \%\) over the same decade, for example, the predicted 2021 debt-to-GDP ratio is \(66 \%\) and the predicted 2021 deficit is \(\$ 521\) billion. If the growth rate is even stronger, at \(3.9 \%,\) the predicted 2021 debt-to-GDP ratio is \(55 \%\) and the predicted 2021 deficit is \(\$ 113\) billion. \(^{71}\) (a) There are only three individual cases given (for three different economic scenarios), and for each we are given values of three variables. What are the variables? (b) Use technology and the three cases given to find the regression line for predicting 2021 debt-toGDP ratio from the average growth rate over the decade 2011 to 2021 . (c) Interpret the slope and intercept of the line from part (b) in context. (d) What 2021 debt-to-GDP ratio does the model in part (b) predict if growth is \(2 \% ?\) If it is \(4 \%\) ? (e) Studies indicate that a country's economic growth slows if the debt-to-GDP ratio hits \(90 \%\). Using the model from part (b), at what growth rate would we expect the ratio in the US to hit \(90 \%\) in \(2021 ?\) (f) Use technology and the three cases given to find the regression line for predicting the deficit (in billions of dollars) in 2021 from the average growth rate over the decade 2011 to 2021 . (g) Interpret the slope and intercept of the line from part (f) in context. (h) What 2021 deficit does the model in part (f) predict if growth is \(2 \% ?\) If it is \(4 \% ?\) (i) The deficit in 2011 was \(\$ 1.4\) trillion. What growth rate would leave the deficit at this level in \(2021 ?\)

In the book Scorecasting, \(^{9}\) we learn that "Across 43 professional soccer leagues in 24 different countries spanning Europe, South America, Asia, Africa, Australia, and the United States (covering more than 66,000 games), the home field advantage [percent of games won by the home team] in soccer worldwide is \(62.4 \% . "\) Is this a population or a sample? What are the cases and approximately how many are there? What is the variable and is it categorical or quantitative? What is the relevant statistic, including correct notation?

Near-Death Experiences People who have a brush with death occasionally report experiencing a near-death experience, which includes the sensation of seeing a bright light or feeling separated from one's body or sensing time speeding up or slowing down. Researchers \(^{14}\) interviewed 1595 people admitted to a hospital cardiac care unit during a recent 30 -month period. Patients were classified as cardiac arrest patients (in which the heart briefly stops after beating unusually quickly) or patients suffering other serious heart problems (such as heart attacks). The study found that 27 individuals reported having had a near-death experience, including 11 of the 116 cardiac arrest patients. Make a two-way table of these data. Compute the appropriate percentages to compare the rate of near-death experiences between the two groups. Describe the results.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.