/*! 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 24 Give two sets of five numbers th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 24

Give two sets of five numbers that have the same mean but different standard deviations, and give two sets of five numbers that have the same standard deviation but different means.

Short Answer

Expert verified
Two sets of five numbers with the same mean but different standard deviations are [3, 3, 3, 3, 3] and [1, 2, 3, 4, 5]. Two sets of five numbers with the same standard deviation but different means are [2, 3, 4, 5, 6] and [12, 13, 14, 15, 16].

Step by step solution

01

Define Sets with Same Mean and Different Standard Deviations

To achieve this, one should pick a common mean and then distribute the numbers around that mean differently for each set. For example, let's take the mean as 3. The first set could be [3, 3, 3, 3, 3], and the second set could be [1, 2, 3, 4, 5]. Both have the same mean but different standard deviations since the numbers in the second set are more spread out.
02

Calculate the Standard Deviations

By calculation, the standard deviation for the first set (using the formula for standard deviation in a sample) is 0 since all the values are equal, and the standard deviation for the second set is approximately 1.58.
03

Define Sets with Same Standard Deviation and Different Means

To get this, one can first pick a common standard deviation and then create sets where the numbers are as spread out from different means. For example, we could keep the standard deviation as 1.58 and then the numbers could be as spread out from the mean. The first set could be [2, 3, 4, 5, 6] and the second set [12, 13, 14, 15, 16]. The numbers are as spread from the mean, but the central value is different.
04

Calculate the Means

By calculation, the mean for the first set is 4 and for the second set is 14. Hence, they have different means but the same standard deviation.

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 mean, often referred to as the average, is a central measure in statistics. To calculate the mean of a data set, sum up all the values and then divide by the total number of values. This gives a central value that represents the typical amount in your set.
For instance, in the data set [1, 2, 3, 4, 5], the mean is calculated as follows:
  • Add up all the numbers: 1 + 2 + 3 + 4 + 5 = 15
  • Divide by the total count of numbers: 15 divided by 5 equals 3
So, the mean of this data set is 3, offering a simple glimpse into where values tend to cluster.
The mean, however, doesn't express the spread of the data, which is where the standard deviation comes into play.
Standard Deviation
Standard deviation is a measure of how spread out the numbers in a data set are from the mean. It provides insight into the variability present in your data.
Let鈥檚 consider two sets with the same mean but different spreads: [3, 3, 3, 3, 3] and [1, 2, 3, 4, 5].
  • For the first set, since all numbers are 3, the numbers do not deviate from the mean, resulting in a standard deviation of 0.
  • In the second set, numbers vary around the mean, leading to a larger standard deviation.
This illustrates how standard deviation can stay the same across different data sets with similar spreads despite contrasting means.
Data Sets
Data sets are collections of numbers or values that you analyze statistically. They can reveal trends and patterns to better understand the data.
For example, to create two data sets with the same standard deviation but different means, consider these sets:
  • [2, 3, 4, 5, 6]
  • [12, 13, 14, 15, 16]
Although both sets have a different central value, they share a similar spread. This helps us see how different data sets can maintain the same measure of variability even while the central tendency shifts.
Variability
Variability describes how data points differ from each other and from the mean. It's crucial for understanding the reliability and predictability of your data.
In our exercise, the key was to observe data sets with differing levels and patterns of variability. Variability can give us a better idea of data consistency. A low variability means data points are close to the mean, as in [3, 3, 3, 3, 3].
On the other hand, a higher variability in a set like [1, 2, 3, 4, 5] indicates more diversity in data points, impacting decisions and predictions made from the data.
Understanding variability alongside the mean and standard deviation is essential in the field of descriptive statistics.

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

The Insurance Institute for Highway Safety (www.iihs.org, June 11,2009 ) published data on repair costs for cars involved in different types of accidents. In one study, seven different 2009 models of mini- and micro-cars were driven at 6 mph straight into a fixed barrier. The following table gives the cost of repairing damage to the bumper for each of the seven models: $$ \begin{array}{lc} \text { Model } & \text { Repair Cost } \\ \hline \text { Smart Fortwo } & \$ 1,480 \\ \text { Chevrolet Aveo } & \$ 1,071 \\ \text { Mini Cooper } & \$ 2,291 \\ \text { Toyota Yaris } & \$ 1,688 \\ \text { Honda Fit } & \$ 1,124 \\ \text { Hyundai Accent } & \$ 3,476 \\ \text { Kia Rio } & \$ 3,701 \\ \hline \end{array} $$ a. Compute the values of the variance and standard deviation. The standard deviation is fairly large. What does this tell you about the repair costs? b. The Insurance Institute for Highway Safety (referenced in the previous exercise) also gave bumper repair costs in a study of six models of minivans (December 30,2007 ). Write a few sentences describing how mini- and micro-cars and minivans differ with respect to typical bumper repair cost and bumper repair cost variability. $$ \begin{array}{lr} \text { Model } & \text { Repair Cost } \\ \hline \text { Honda Odyssey } & \$ 1,538 \\ \text { Dodge Grand Caravan } & \$ 1,347 \\ \text { Toyota Sienna } & \$ 840 \\ \text { Chevrolet Uplander } & \$ 1,631 \\ \text { Kia Sedona } & \$ 1,176 \\ \text { Nissan Quest } & \$ 1,603 \\ \hline \end{array} $$

Consider the following statement: More than \(65 \%\) of the residents of Los Angeles earn less than the average wage for that city. Could this statement be correct? If so, how? If not, why not?

The accompanying data on number of minutes used for cell phone calls in one month was generated to be consistent with summary statistics published in a report of a marketing study of San Diego residents (TeleTruth, March 2009 ): $$ \begin{array}{rrrrrrrrrr} 189 & 0 & 189 & 177 & 106 & 201 & 0 & 212 & 0 & 306 \\ 0 & 0 & 59 & 224 & 0 & 189 & 142 & 83 & 71 & 165 \\ 236 & 0 & 142 & 236 & 130 & & & & & \end{array} $$ a. Would you recommend the mean or the median as a measure of center for this data set? Give a brief explanation of your choice. (Hint: It may help to look at a graphical display of the data.) b. Compute a trimmed mean by deleting the three smallest observations and the three largest observations in the data set and then averaging the remaining 19 observations. What is the trimming percentage for this trimmed mean? c. What trimming percentage would you need to use in order to delete all of the 0 minute values from the data set? Would you recommend a trimmed mean with this trimming percentage? Explain why or why not.

An instructor has graded 19 exam papers submitted by students in a class of 20 students, and the average so far is 70 . (The maximum possible score is \(100 .\) ) How high would the score on the last paper have to be to raise the class average by 1 point? By 2 points?

The following data are cost (in cents) per ounce for nine different brands of sliced Swiss cheese (www .consumerreports.org): \(\begin{array}{llllllll}29 & 62 & 37 & 41 & 70 & 82 & 47 & 52\end{array}\) a. Compute the variance and standard deviation for this data set. \(s^{2}=279.111 ; s=16.707\) 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?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure 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.