/*! 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 13 In Problem 25 from Section 3.1, ... [FREE SOLUTION] | 91Ó°ÊÓ

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In Problem 25 from Section 3.1, we drew a histogram of the weights of \(\mathrm{M\&Ms}\) and found that the distribution is symmetric. Draw a box plot of these data. Use the box plot and quartiles to confirm the distribution is symmetric. For convenience, the data are displayed again. $$ \begin{array}{ccccccc} \hline 0.87 & 0.88 & 0.82 & 0.90 & 0.90 & 0.84 & 0.84 \\ \hline 0.91 & 0.94 & 0.86 & 0.86 & 0.86 & 0.88 & 0.87 \\ \hline 0.89 & 0.91 & 0.86 & 0.87 & 0.93 & 0.88 & \\ \hline 0.83 & 0.95 & 0.87 & 0.93 & 0.91 & 0.85 & \\ \hline 0.91 & 0.91 & 0.86 & 0.89 & 0.87 & 0.84 & \\ \hline 0.88 & 0.88 & 0.89 & 0.79 & 0.82 & 0.83 & \\ \hline 0.90 & 0.88 & 0.84 & 0.93 & 0.81 & 0.90 & \\ \hline 0.88 & 0.92 & 0.85 & 0.84 & 0.84 & 0.86 & \\ \hline \end{array} $$

Short Answer

Expert verified
The box plot confirms the distribution is symmetric because the box plot is even and balanced.

Step by step solution

01

Arrange the Data in Ascending Order

First, arrange the given data in ascending order. The sorted data is: 0.79, 0.81, 0.82, 0.82, 0.83, 0.83, 0.84, 0.84, 0.84, 0.84, 0.84, 0.85, 0.85, 0.86, 0.86, 0.86, 0.86, 0.86, 0.87, 0.87, 0.87, 0.87, 0.87, 0.88, 0.88, 0.88, 0.88, 0.88, 0.88, 0.89, 0.89, 0.89, 0.90, 0.90, 0.90, 0.90, 0.91, 0.91, 0.91, 0.91, 0.91, 0.92, 0.93, 0.93, 0.93, 0.94, 0.95
02

Find the Quartiles

Divide the data into four equal parts to find the first quartile (Q1), second quartile (Q2 or median), and third quartile (Q3).Q1 (first quartile): 0.84Q2 (median or second quartile): 0.88Q3 (third quartile): 0.90
03

Determine the Min and Max Values

Identify the minimum and maximum values in the data set.Min value: 0.79Max value: 0.95
04

Draw the Box Plot

Using the quartiles and the minimum and maximum values, draw a box plot:- Draw a number line that fits the range of your data (min to max).- Draw a rectangle (the ‘box’) from Q1 to Q3.- Inside the box, draw a line at Q2 (median).- Draw 'whiskers' from the box to the min and max values.
05

Check for Symmetry

Examine the box plot. The distribution is symmetric if the left and right sides of the box are approximately equal, and the whiskers are about the same length on both sides.

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Key Concepts

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

quartiles
Quartiles are key values that divide your data into four equal parts. They help us summarize data and understand its distribution. The first quartile (Q1) is the median of the lower half of the data, the second quartile (Q2) is the median of the entire dataset, and the third quartile (Q3) is the median of the upper half of the data.
In the given dataset of M&Ms weights, we have organized the data in ascending order. We found:
  • Q1: 0.84
  • Q2 (median): 0.88
  • Q3: 0.90
The distance between these quartiles helps us understand the spread and center of our data effectively.
symmetric distribution
A symmetric distribution means that the left side of the data is a mirror image of the right side. In a box plot, this symmetry will be evident if the box and whiskers are equally spread out on both sides of the median.
Looking at our given data:
  • Min value: 0.79
  • Q1: 0.84
  • Median: 0.88
  • Q3: 0.90
  • Max value: 0.95
In this dataset, the distances between Q1 and the median, and the median and Q3, are relatively equal. Similarly, the whiskers (lines extending from the box) are also equitable on both sides, indicating a symmetric distribution.
histogram
A histogram is a graphical representation of data. It shows the frequency of data points within certain ranges, helping us visualize the shape of the data distribution.
When creating a histogram for the M&Ms weights, we can see if the distribution is symmetric.
Frequently, a symmetric histogram will have a bell-shaped curve. Reviewing our data, the histogram showed a peak at the center, with the frequencies dropping off equally on both sides, confirming the symmetric nature of the distribution. This was a foundational step before using the box plot for further analysis.
data visualization
Data visualization turns complex numerical data into comprehensive visual formats like charts, graphs, and plots. It helps in quickly understanding trends, outliers, and patterns.
In this exercise, we utilized two main visualization tools: the histogram and the box plot. The histogram gave us an overview of the frequency distribution, indicating symmetry.
The box plot further confirmed symmetry by displaying the quartiles, median, minimum, and maximum clearly. Visualizations help us to get an intuitive grasp of data and make more informed decisions based on our observations.
descriptive statistics
Descriptive statistics summarize data to provide a clear view of its basic features. It includes measures like mean, median, mode, range, and quartiles.
In our dataset, using descriptive statistics, we calculated:
  • Minimum: 0.79
  • Q1 (first quartile): 0.84
  • Median (Q2): 0.88
  • Q3 (third quartile): 0.90
  • Maximum: 0.95
These statistics summarized the entire dataset into key points, making it easier to understand the distribution. Using descriptive statistics aids in interpreting data without going through every data point individually.

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Most popular questions from this chapter

The following data represent the pulse rates (beats per minute) of nine students enrolled in a section of Sullivan's Introductory Statistics course. Treat the nine students as a population. $$ \begin{array}{lc} \text { Student } & \text { Pulse } \\ \hline \text { Perpectual Bempah } & 76 \\ \hline \text { Megan Brooks } & 60 \\ \hline \text { Jeff Honeycutt } & 60 \\ \hline \text { Clarice Jefferson } & 81 \\ \hline \text { Crystal Kurtenbach } & 72 \\ \hline \text { Janette Lantka } & 80 \\ \hline \text { Kevin McCarthy } & 80 \\ \hline \text { Tammy Ohm } & 68 \\ \hline \text { Kathy Wojdyla } & 73 \\ \hline \end{array} $$ (a) Determine the population mean pulse. (b) Find three simple random samples of size 3 and determine the sample mean pulse of each sample. (c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Explain the meaning of the following percentiles. Source: Advance Data from Vital and Health Statistics (a) The 15 th percentile of the head circumference of males 3 to 5 months of age is \(41.0 \mathrm{~cm}\) (b) The 90 th percentile of the waist circumference of females 2 years of age is \(52.7 \mathrm{~cm}\) (c) Anthropometry involves the measurement of the human body. One goal of these measurements is to assess how body measurements may be changing over time. The following table represents the standing height of males aged 20 years or older for various age groups. Based on the percentile measurements of the different age groups, what might you conclude? $$ \begin{array}{llllll} & {\text { Percentile }} \\\\\hline { } \text { Age } & \text { 10th } & \text { 25th } & \text { 50th } & \text { 75th } & \text { 90th } \\ \hline 20-29 & 166.8 & 171.5 & 176.7 & 181.4 & 186.8 \\ \hline 30-39 & 166.9 & 171.3 & 176.0 & 181.9 & 186.2 \\ \hline 40-49 & 167.9 & 172.1 & 176.9 & 182.1 & 186.0 \\ \hline 50-59 & 166.0 & 170.8 & 176.0 & 181.2 & 185.4 \\ \hline 60-69 & 165.3 & 170.1 & 175.1 & 179.5 & 183.7 \\ \hline 70-79 & 163.2 & 167.5 & 172.9 & 178.1 & 181.7 \\ \hline 80 \text { or older } & 161.7 & 166.1 & 170.5 & 175.3 & 179.4\\\ \hline \end{array} $$

As part of its "Customers First" program, a cellular phone company monitors monthly phone usage. The program identifies unusual use and alerts the customer that their phone may have been used by another person. The data below represent the monthly phone use in minutes of a customer enrolled in this program for the past 20 months. The phone company decides to use the upper fence as the cutoff point for the number of minutes at which the customer should be contacted. What is the cutoff point? $$ \begin{array}{rrrrr} \hline 346 & 345 & 489 & 358 & 471 \\ \hline 442 & 466 & 505 & 466 & 372 \\ \hline 442 & 461 & 515 & 549 & 437 \\ \hline 480 & 490 & 429 & 470 & 516 \\ \hline \end{array} $$

Does hunger improve strategic decision making? That is, if you are hungry are you more likely to make a favorable decision when the outcome of your decision is uncertain (as in business decisions)? To test this theory, researchers randomly divided 30 normal weight individuals into two groups. All subjects were asked to refrain from eating or drinking (except water) from 11 P.M. on the day prior to their 9 A.M. meeting. At 9 A.M., the subjects were randomly assigned to one of two groups. The subjects in Group 1 were fed breakfast while the subjects in Group 2 were not fed. All subjects were administered a computerized version of an exam that assesses complex decision making under uncertain conditions. The assessment consisted of subjects choosing cards from four decks marked \(\mathrm{A}, \mathrm{B}, \mathrm{C},\) and D. Cards in decks \(A\) and \(B\) had a point value of 100 while cards in decks \(C\) and \(D\) had point values of \(50 .\) However, deck A had penalty cards that deducted points between 150 and \(300 ;\) deck \(B\) had one penalty card of \(1250 ;\) deck \(C\) had penalty cards between 25 and 75 points; deck D had a single penalty card of 250 points. So, decks A and B had stiffer penalties over the long haul than decks \(C\) and \(D\) and in the long haul, decks \(C\) and \(D\) resulted in more points than decks \(A\) and \(B\). In total, the subjects would select 100 cards. However, the response variable was the number of cards selected from decks \(\mathrm{C}\) and \(\mathrm{D}\) out of the last 60 cards selected. The thinking here is that after 40 card selections, the subjects would be aware of the advantage of decks \(\mathrm{C}\) and \(\mathrm{D}\). The researchers administered a Barret Impulsivity Scale to be sure the two groups did not differ in terms of impulsivity (e.g., "I do things without thinking"). There was no difference in impulsivity, age, or body mass index between the two groups. Before the exam, subjects were asked to report their level of hunger and it was found that Group 2 was significantly more hungry than Group \(1 .\) After analysis of the data, it was determined that the mean number of advantageous cards (decks \(C\) and \(D\) ) selected by the subjects in Group 2 was 33.36 cards while the mean was 25.86 for the subjects in Group \(1 .\) The researchers concluded that hunger improves advantageous decision making. Source: de Ridder, D., Kroese, F., Adriaanse, M., \& Evers, C. "Always Gamble on an Empty Stomach: Hunger Is Associated with Advantageous Decision Making," PLOS One \(9(10) .\) doi: 10.1371 /journal. pone. \(0111081 .\) (a) What type of experimental design is this? (b) Identify the experimental units. (c) What is the response variable? Is it qualitative or quantitative? If quantitative, is it discrete or continuous? (d) What factors that might impact the response variable are cited in the article? Which factor is manipulated? How many levels are there for the manipulated factor? (e) What role does randomization play in the study? How do the researchers verify that randomization resulted in similar groups prior to the treatment? (f) What are the statistics in the study? (g) What is the conclusion of the study?

The following data represent exam scores in a statistics class taught using traditional lecture and a class taught using a "flipped" classroom. The "flipped" classroom is one where the content is delivered via video and watched at home, while class time is used for homework and activities. $$ \begin{array}{llllllll} \hline \text { Traditional } & 70.8 & 69.1 & 79.4 & 67.6 & 85.3 & 78.2 & 56.2 \\\ & 81.3 & 80.9 & 71.5 & 63.7 & 69.8 & 59.8 & \\ \hline \text { Fipped } & 76.4 & 71.6 & 63.4 & 72.4 & 77.9 & 91.8 & 78.9 \\ & 76.8 & 82.1 & 70.2 & 91.5 & 77.8 & 76.5 & \end{array} $$ (a) Which course has more dispersion in exam scores using the range as the measure of dispersion? (b) Which course has more dispersion in exam scores using the standard deviation as the measure of dispersion? (c) Suppose the score of 59.8 in the traditional course was incorrectly recorded as \(598 .\) How does this affect the range? the standard deviation? What property does this illustrate?

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