/*! 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 17 The following data are cost (in ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 17

The following data are cost (in cents) per ounce for nine different brands of sliced Swiss cheese (www .consumerreports.org): \(\begin{array}{lllllllll}29 & 62 & 37 & 41 & 70 & 82 & 47 & 52 & 49\end{array}\) a. Compute the variance and standard deviation for this data set. b. If a very expensive cheese with a cost per slice of 150 cents was added to the data set, how would the values of the mean and standard deviation change?

Short Answer

Expert verified
The variance and standard deviation for the given data set are approximately 304.1 and 17.4 respectively. When a new element (150) is added, the mean becomes 61.7 and the new standard deviation is around 30.5.

Step by step solution

01

- Calculate the Mean

Firstly, take all values and calculate their mean (also known as average). This can be done by adding all values together and dividing by the total number of values. Here, for these set of values (29, 62, 37, 41, 70, 82, 47, 52, 49), the sum is 469 and the number of values or 'n' is 9, which gives a mean of 52.1.
02

- Calculate the Variance

Next, subtract the mean from each value in the set, then square the result. This gives the 'squared deviation'. Add up all of these squared deviations, then divide by 'n-1' (the number of values minus 1). In this case, 'n-1' would be 8. After calculating, we find the variance to be 304.1.
03

- Calculate the Standard Deviation

To derive the standard deviation, the square root of the variance is taken. After computing, we get the standard deviation to be approximately 17.4
04

- Analyze the Mean after Adding a New Element

Adding a new value into the dataset will obviously affect the mean. Here, add 150 into the set and recalculate the mean. This results in the new mean to be approximately 61.7.
05

- Analyze the Standard Deviation after Adding a New Element

Similarly, recalculate the standard deviation after reintroducing the new element into the dataset. After recalculating, the new standard deviation is found to be approximately 30.5

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.

Variance Calculation
The concept of variance in statistics is fundamental for understanding the spread of data points around the mean. Variance tells us how much the data points in a set differ from the mean, providing insights into the variability within the dataset.

To calculate variance, follow these steps:
  • First, determine the mean of the dataset, which is the average value of all data points.
  • Then subtract the mean from each individual data point. This is known as the 'deviation' from the mean for each data point.
  • After finding the deviation for each data point, square these deviations. The purpose of squaring is to eliminate negative numbers and to give more weight to larger differences.
  • Sum all the squared deviations.
  • Finally, divide this sum by one less than the number of data points (\( n-1 \)). This step corrects for bias in the estimation of the population variance.
In our example featuring nine brands of Swiss cheese, after conducting these calculations, we found that the variance was approximately 304.1.
Standard Deviation
The standard deviation is a metric that expresses how much variation or "dispersion" exists from the average (mean), indicating the typical deviation of the data points. It is derived directly from the variance and provides an intuitive measure of variability.

Calculating the standard deviation involves these steps:
  • Compute the variance first, as previously described. Variance represents the squared deviations from the mean.
  • Take the square root of the variance. The square root is used to return the unit of measure to the same as the original dataset and make the measure of spread easier to interpret.
In the Swiss cheese cost example, the computed standard deviation is roughly 17.4. This means, on average, the costs deviate from the mean of 52.1 cents by around 17.4 cents. After adding an expensive cheese priced at 150 cents, this significantly increases the variability, reflecting in a new standard deviation of approximately 30.5.
Mean Calculation
The mean, often referred to as the average, is a central measure of tendency. It provides a quick snapshot of typical values in a dataset and serves as a basis for many statistical analyses.

The mean is calculated in a straightforward manner:
  • Add together all the values in the dataset. This summation gives the total aggregate of the dataset.
  • Divide this sum by the total number of values, known as the sample size (\( n \)). This gives you the average value per unit in the dataset.
In the case of the cheese example, the mean was initially 52.1 cents per ounce, calculated by adding up the costs and dividing by nine, the number of brands. Upon including the cost of a higher-priced cheese, the mean increases, moving to approximately 61.7. This shift illustrates how the average is influenced by extreme values in the dataset.
Data Analysis
Data analysis involves a comprehensive process of examining, cleaning, and modeling data to understand outcomes and make informed decisions. It is pivotal in research, business modeling, and virtually any field involving quantitative data.

Data analysis includes the following stages:
  • Collecting Data: Gathering the relevant data is the first step, ensuring it is accurate and complete.
  • Organizing Data: Data needs to be organized in a digestible format, often through tables, charts, or graphs, to identify patterns or trends.
  • Descriptive Analysis: This involves calculating measures like the mean, variance, and standard deviation to summarize and interpret data.
  • Inferential Analysis: Using statistical techniques to draw conclusions based on the data and make predictions.
  • Decision Making: Based on insights from the data analysis, informed decisions can be made, be it in business strategies or scientific research.
In our cheese example, data analysis helped us understand the distribution and variability of cheese costs, identify outliers, and predict the impact of price changes on overall cheese cost metrics.

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

A sample of concrete specimens of a certain type is selected, and the compressive strength of each specimen is determined. The mean and standard deviation are calculated as \(\bar{x}=3000\) and \(s=500\), and the sample histogram is found to be well approximated by a normal curve. a. Approximately what percentage of the sample observations are between 2500 and 3500 ? b. Approximately what percentage of sample observations are outside the interval from 2000 to 4000 ? c. What can be said about the approximate percentage of observations between 2000 and \(2500 ?\) d. Why would you not use Chebyshev's Rule to answer the questions posed in Parts (a)-(c)?

The risk of developing iron deficiency is especially high during pregnancy. Detecting such a deficiency is complicated by the fact that some methods for determining iron status can be affected by the state of pregnancy itself. Consider the following data on transferrin receptor concentration for a sample of women with laboratory evidence of overt iron-deficiency anemia ("Serum Transferrin Receptor for the Detection of Iron Deficiency in Pregnancy," American journal of Clinical Nutrition [1991]: \(1077-1081\) ): $$ \begin{array}{llrlrl} 15.2 & 9.3 & 7.6 & 11.9 & 10.4 & 9.7 \\ 20.4 & 9.4 & 11.5 & 16.2 & 9.4 & 8.3 \end{array} $$ Compute the values of the sample mean and median. Why are these values different here? Which one do you regard as more representative of the sample, and why?

The paper "Portable Sodal Groups: Willingness to Communicate, Interpersonal Communication Gratifications, and Cell Phone Use among Young Adults" (International journal of Mobile Communications [2007]: \(139-156\) ) describes a study of young adult cell phone use patterns. a. Comment on the following quote from the paper. Do you agree with the authors?? Seven sections of an Introduction to Mass Communication course at a large southern university were surveyed in the spring and fall of 2003 . The sample was chosen because it offered an excellent representation of the population under studyyoung adults. b. Below is another quote from the paper. In this quote, the author reports the mean number of minutes of cell phone use per week for those who participated in the survey. What additional information would have been provided about cell phone use behavior if the author had also reported the standard deviation? Based on respondent estimates, users spent an average of 629 minutes (about \(10.5\) hours) per week using their cell phone on or off line for any reason.

In 1997, a woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard (Genessey v. Digital Equipment Corporation). The jury awarded about \(\$ 3.5\) million for pain and suffering, but the court then set aside that award as being unreasonable compensation. In making this determination, the court identified a "normative" group of 27 similar cases and specified a reasonable award as one within 2 standard deviations of the mean of the awards in the 27 cases. The 27 award amounts were (in thousands of dollars) \(\begin{array}{rrrrrrrr}37 & 60 & 75 & 115 & 135 & 140 & 149 & 150 \\ 238 & 290 & 340 & 410 & 600 & 750 & 750 & 750 \\\ 1050 & 1100 & 1139 & 1150 & 1200 & 1200 & 1250 & 1576 \\ 1700 & 1825 & 2000 & & & & & \end{array}\) What is the maximum possible amount that could be awarded under the "2-standard deviations rule?"

The article "Taxable Wealth and Alcoholic Beverage Consumption in the United States" (Psychological Reports [1994]: \(813-814\) ) reported that the mean annual adult consumption of wine was \(3.15\) gallons and that the standard deviation was \(6.09\) gallons. Would you use the Empirical Rule to approximate the proportion of adults who consume more than \(9.24\) gallons (i.e., the proportion of adults whose consumption value exceeds the mean by more than 1 standard deviation)? Explain your reasoning.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.