/*! 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 10 Consider two data sets. Set \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 10

Consider two data sets. Set \(\mathrm{A}: n=5 ; \bar{x}=10 \quad\) Set B: \(n=50 ; \bar{x}=10\) (a) Suppose the number 20 is included as an additional data value in Set \(\mathrm{A}\). Compute \(\bar{x}\) for the new data set. Hint: \(\Sigma x=n \bar{x} .\) To compute \(\bar{x}\) for the new data set, add 20 to \(\Sigma x\) of the original data set and divide by 6 . (b) Suppose the number 20 is included as an additional data value in Set \(\mathrm{B}\). Compute \(\bar{x}\) for the new data set. (c) Why does the addition of the number 20 to each data set change the mean for Set A more than it does for Set B?

Short Answer

Expert verified
The new mean of Set A is 11.67 and Set B is 10.20. The mean changes more for Set A because it has fewer data points than Set B.

Step by step solution

01

Determine the Sum of Set A

To find the new mean, we first calculate the current sum of Set A using the formula \( \Sigma x = n \bar{x} \). For Set A, \( n = 5 \) and \( \bar{x} = 10 \). Thus, the sum \( \Sigma x = 5 \times 10 = 50 \).
02

Add 20 to the Sum of Set A and Recalculate Mean

Next, add 20 to the sum previously calculated. The new sum is \( 50 + 20 = 70 \). Since the number of values is now 6, the new mean is calculated as \( \bar{x} = \frac{70}{6} \approx 11.67 \).
03

Determine the Sum of Set B

For Set B, use the same approach to determine the sum. Given \( n = 50 \) and \( \bar{x} = 10 \), the sum is \( \Sigma x = 50 \times 10 = 500 \).
04

Add 20 to the Sum of Set B and Recalculate Mean

Add 20 to the existing sum. The new sum is \( 500 + 20 = 520 \). Calculate the mean with the new count of 51, which results in \( \bar{x} = \frac{520}{51} \approx 10.20 \).
05

Compare the Impact on Means for Both Sets

The mean of Set A increased from 10 to approximately 11.67, while the mean of Set B increased slightly from 10 to approximately 10.20. The larger the sample size, the less impact an additional data value has on the mean. This is why the mean of Set A changes more significantly than Set B.

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
The concept of the mean, often called the average, is a crucial aspect of descriptive statistics.
It helps us understand a central tendency of a data set.To calculate the mean, you sum up all data values in your set and then divide that total by the number of data points. Mathematically, it's expressed as:\[\bar{x} = \frac{\Sigma x}{n}\]Where:
  • \(\bar{x}\) is the mean.
  • \(\Sigma x\) represents the sum of all data points.
  • \(n\) is the number of data points.
Understanding the mean is essential for comparing data sets. As seen in the exercise, when a new data value like "20" is added to Sets A and B, it increases the total sum, thereby affecting the mean.
The mean indicates a central value where the data points tend to cluster and can be easily influenced by singular high or low values, especially in smaller sets.
Sample Size
Sample size, represented by \(n\), is the total number of observations or data points in a data set.
It鈥檚 a fundamental component in statistics that influences the reliability and precision of statistical measures, such as the mean.In the provided exercise, Set A has a sample size of 5, whereas Set B has 50.
This difference plays an integral role in understanding how new data points impact each set's mean.When the number 20 is added as an additional data value:
  • The impact on the mean of Set A is significant because the sample size is small.
  • The impact on Set B is minimal as its larger sample size "dilutes" the effect of this single additional value.
The larger a sample size becomes, the less susceptible the data's statistical measures are to changes by any one data point.
This highlights the importance of considering sample size when interpreting data.
Data Set
A data set is a collection of numbers or values that represent some information or measurements.
They are the foundation upon which statistical analyses and interpretations are based. Data sets can vary widely in size, composition, and form. In statistics, data sets are often used to calculate measures such as mean, median, mode, standard deviation, etc.
Each measure helps offer different insights into the characteristics of the data. For example, in the exercise:
  • Set A has a mean of 10 before adding any new value and consists of 5 data points.
  • Set B, having the same mean to start with, comprises 50 data points.
When examining data sets:
  • Consider how the number and value of data points influence overall calculations and conclusions.
  • Focus on the distribution of data points and how changes impact statistical measures.
Understanding a data set's structure and properties helps make logical and meaningful analysis, as evident when adding the number 20 to both sets and observing the varying effects on their respective means.

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

Kevlar epoxy is a material used on the NASA space shuttles. Strands of this epoxy were tested at the \(90 \%\) breaking strength. The following data represent time to failure (in hours) for a random sample of 50 epoxy strands (Reference: R. E. Barlow, University of California, Berkeley). Let \(x\) be a random variable representing time to failure (in hours) at \(90 \%\) breaking strength. Note: These data are also available for download at the Online Study Center. \(\begin{array}{llllllllll}0.54 & 1.80 & 1.52 & 2.05 & 1.03 & 1.18 & 0.80 & 1.33 & 1.29 & 1.11 \\ 3.34 & 1.54 & 0.08 & 0.12 & 0.60 & 0.72 & 0.92 & 1.05 & 1.43 & 3.03 \\ 1.81 & 2.17 & 0.63 & 0.56 & 0.03 & 0.09 & 0.18 & 0.34 & 1.51 & 1.45 \\ 1.52 & 0.19 & 1.55 & 0.02 & 0.07 & 0.65 & 0.40 & 0.24 & 1.51 & 1.45 \\ 1.60 & 1.80 & 4.69 & 0.08 & 7.89 & 1.58 & 1.64 & 0.03 & 0.23 & 0.72\end{array}\) (a) Find the range. (b) Use a calculator to verify that \(\Sigma x=62.11\) and \(\Sigma x^{2} \approx 164.23\). (c) Use the results of part (b) to compute the sample mean, variance, and standard deviation for the time to failure. (d) Use the results of part (c) to compute the coefficient of variation. What does this number say about time to failure? Why does a small \(C V\) indicate more consistent data, whereas a larger \(\mathrm{CV}\) indicates less consistent data? Explain.

Consider the following types of data that were obtained from a random sample of 49 credit card accounts. Identify all the averages (mean, median, or mode) that can be used to summarize the data. (a) Outstanding balance on each account (b) Name of credit card (e.g., MasterCard, Visa, American Express, etc.) (c) Dollar amount due on next payment

What symbol is used for the arithmetic mean when it is a sample statistic? What symbol is used when the arithmetic mean is a population parameter?

Some data sets include values so high or so low that they seem to stand apart from the rest of the data. These data are called outliers. Outliers may represent data collection errors, data entry errors, or simply valid but unusual data values. It is important to identify outliers in the data set and examine the outliers carefully to determine if they are in error. One way to detect outliers is to use a box-and-whisker plot. Data values that fall beyond the limits Lower limit: \(Q_{1}-1.5 \times(I Q R)\) Upper limit: \(Q_{3}+1.5 \times(I Q R)\) where \(I Q R\) is the interquartile range, are suspected outliers. In the computer software package Minitab, values beyond these limits are plotted with asterisks ( \(^{*}\) ). Students from a statistics class were asked to record their heights in inches. The heights (as recorded) were \(\begin{array}{llllllllllll}65 & 72 & 68 & 64 & 60 & 55 & 73 & 71 & 52 & 63 & 61 & 74 \\ 69 & 67 & 74 & 50 & 4 & 75 & 67 & 62 & 66 & 80 & 64 & 65\end{array}\) (a) Make a box-and-whisker plot of the data. (b) Find the value of the interquartile range \((I Q R)\). (c) Multiply the \(I Q R\) by \(1.5\) and find the lower and upper limits. (d) Are there any data values below the lower limit? above the upper limit? List any suspected outliers. What might be some explanations for the outliers?

Given the sample data \(\begin{array}{llllll}x: & 23 & 17 & 15 & 30 & 25\end{array}\) (a) Find the range. (b) Verify that \(\Sigma x=110\) and \(\Sigma x^{2}=2568\). (c) Use the results of part (b) and appropriate computation formulas to compute the sample variance \(s^{2}\) and sample standard deviation \(s\). (d) Use the defining formulas to compute the sample variance \(s^{2}\) and sample standard deviation \(s\). (e) Suppose the given data comprise the entire population of all \(x\) values. Compute the population variance \(\sigma^{2}\) and population standard deviation \(\sigma\).
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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.