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Chapter 3: Problem 36

The average college student produces 640 pounds of solid waste each year. If the standard deviation is approximately 85 pounds, within what weight limits will at least \(88.89 \%\) of all students garbage lie?

Short Answer

Expert verified
The weights will lie between 385 and 895 pounds.

Step by step solution

01

Understand the Problem

We are given that the average amount of garbage produced by a college student is 640 pounds with a standard deviation of 85 pounds. We need to find the range within which at least 88.89% of students' garbage will lie.
02

Apply the Empirical Rule

The Empirical Rule, or 68-95-99.7 rule, isn't exact beyond three standard deviations. For 88.89% precision, Chebyshev's Theorem, applicable for all distributions, can help us determine this range effectively.
03

Use Chebyshev's Theorem

Chebyshev鈥檚 Theorem states that for any dataset and number of standard deviations, at least \( 1 - \frac{1}{k^2} \) of the data falls within \( k \) standard deviations of the mean. To cover 88.89%: \[ 1 - \frac{1}{k^2} = 0.8889 \] Solving for \( k \), \[ \frac{1}{k^2} = 0.1111 \] \[ k^2 = \frac{1}{0.1111} \approx 9.0009 \] \[ k \approx 3 \]
04

Calculate the Weight Limits

With \( k \approx 3 \), determine the intervals: Mean = 640 pounds, Standard deviation = 85 pounds, \(640 \pm 3 \times 85\). Calculating this gives:- Lower limit: \(640 - 255 = 385\) pounds- Upper limit: \(640 + 255 = 895\) poundsThus, at least 88.89% of weights will be between 385 and 895 pounds.

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Key Concepts

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

Understanding Standard Deviation
Standard deviation is a way to measure the amount of variation or dispersion in a set of values. Imagine it as a ruler that helps you see how spread out the numbers in your data are from the average, or mean. The standard deviation is crucial because it gives you a tangible sense of variability.
  • A small standard deviation means the data points are close to the mean.
  • A large standard deviation indicates the data points are spread out over a larger range of values.
In our scenario, with a standard deviation of 85 pounds, it tells us that the garbage amounts can vary significantly around the mean of 640 pounds. If you visualize it, imagine a bell curve with the center being the most common value, which is the mean, and the curve's width indicating the amount of variability. This is where the standard deviation comes into play, showing us just how wide that curve is.
Exploring the Empirical Rule
The Empirical Rule is a helpful guideline in statistics that describes how data spreads in a normal distribution. It's often called the 68-95-99.7 rule because it states that:
  • About 68% of data falls within one standard deviation of the mean.
  • Approximately 95% lie within two standard deviations.
  • Roughly 99.7% are within three standard deviations.
While this rule is great for normally distributed data, it has its limitations. Our exercise involves calculating a range that contains 88.89% of the data, which falls outside this handy rule's scope. That鈥檚 when Chebyshev鈥檚 Theorem comes in to provide a broader understanding applicable to any dataset, regardless of distribution shape.
Defining the Mean
The mean, often referred to as the average, is like the central hub of your data, offering a single value to represent a whole set. It鈥檚 calculated by summing all the numbers in a dataset and then dividing by the number of values. This measure tells us where the center of the data lies.
In the context of our problem, the mean is provided as 640 pounds, signifying that on average, each student produces this amount of waste annually. This number provides a benchmark to compare individual and group data points. When we calculate ranges or consider the spread (as with the standard deviation), it's the mean that acts as our anchor point, showing how far and in which direction the points deviate from."

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Most popular questions from this chapter

The data show a sample of the number of passengers in millions that major airlines carried for a recent year. Find the mean, median, midrange, and mode for the data. $$\begin{array}{rrrrrrr}143.8 & 17.7 & 8.5 & 120.4 & 33.0 & 7.1 & 27.1 \\\10.0 & 5.0 & 6.1 & 4.3 & 3.1 & 12.1 &\end{array}$$

Describe which measure of central tendency \(-\) mean, median, or mode-was probably used in each situation. a. One-half of the factory workers make more than \(\$ 5.37\) per hour, and one- half make less than \(\$ 5.37\) per hour. b. The average number of children per family in the Plaza Heights Complex is 1.8 . c. Most people prefer red convertibles over any other color. d. The average person cuts the lawn once a week. e. The most common fear today is fear of speaking in public. f. The average age of college professors is 42.3 years.

A four-month record for the number of tornadoes in \(2013-2015\) is given here. $$\begin{array}{lrrr} & 2013 & 2014 & 2015 \\\\\hline \text { April } & 80 & 130 & 170 \\\\\text { May } & 227 & 133 & 382 \\\\\text { June } & 126 & 280 & 184 \\\\\text { July } & 69 & 90 & 116\end{array}$$ a. Which month had the highest mean number of tornadoes for this 3 -year period? b. Which year has the highest mean number of tornadoes for this 4 -month period? c. Construct three boxplots and compare the distributions.

Team batting averages for major league baseball in 2015 are represented below. Find the variance and standard deviation for each league. Compare the results. $$ \begin{array}{crcc} \text { NL } & & \text { AL } & \\ \hline 0.242-0.246 & 3 & 0.244-0.249 & 3 \\ 0.247-0.251 & 6 & 0.250-0.255 & 6 \\ 0.252-0.256 & 1 & 0.256-0.261 & 2 \\ 0.257-0.261 & 11 & 0.262-0.267 & 1 \\ 0.262-0.266 & 11 & 0.268-0.273 & 3 \\ 0.267-0.271 & 1 & 0.274-0.279 & 0 \end{array} $$

The number of annual precipitation days for one-half of the 50 largest U.S. cities is listed below. Find the range, variance, and standard deviation of the data. \(\begin{array}{rrrrrrrrrr}135 & 128 & 136 & 78 & 116 & 77 & 111 & 79 & 44 & 97 \\\ 116 & 123 & 88 & 102 & 26 & 82 & 156 & 133 & 107 & 35 \\ 112 & 98 & 45 & 122 & 125 & & & & & \end{array}\)
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