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1.3: Presentation of Data

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Skills to Develop

• To learn two ways that data will be presented in the text.

In this book we will use two formats for presenting data sets. The first is a data list, which is an explicit listing of all the individual measurements, either as a display with space between the individual measurements, or in set notation with individual measurements separated by commas.

Example \(\PageIndex{1}\)

The data obtained by measuring the age of \(21\) randomly selected students enrolled in freshman courses at a university could be presented as the data list:

\[\begin{array}{cccccccccc}18 & 18 & 19 & 19 & 19 & 18 & 22 & 20 & 18 & 18 & 17 \\ 19 & 18 & 24 & 18 & 20 & 18 & 21 & 20 & 17 & 19 &\end{array}\]

or in set notation as:

\[ \{18,18,19,19,19,18,22,20,18,18,17,19,18,24,18,20,18,21,20,17,19\} \]

A data set can also be presented by means of a data frequency table, a table in which each distinct value \(x\) is listed in the first row and its frequency \(f\), which is the number of times the value \(x\) appears in the data set, is listed below it in the second row.

Example \(\PageIndex{2}\)

The data set of the previous example is represented by the data frequency table

\[\begin{array}{c|cccccc}x & 17 & 18 & 19 & 20 & 21 & 22 & 24 \\ \hline f & 2 & 8 & 5 & 3 & 1 & 1 & 1\end{array}\]

The data frequency table is especially convenient when data sets are large and the number of distinct values is not too large.

Key Takeaway

• Data sets can be presented either by listing all the elements or by giving a table of values and frequencies.

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1.3: Presentation of Data

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Learning Objectives

• To learn two ways that data will be presented in the text.

In this book we will use two formats for presenting data sets. The first is a data list, which is an explicit listing of all the individual measurements, either as a display with space between the individual measurements, or in set notation with individual measurements separated by commas.

Example \(\PageIndex{1}\)

The data obtained by measuring the age of \(21\) randomly selected students enrolled in freshman courses at a university could be presented as the data list:

\[\begin{array}{cccccccccc}18 & 18 & 19 & 19 & 19 & 18 & 22 & 20 & 18 & 18 & 17 \\ 19 & 18 & 24 & 18 & 20 & 18 & 21 & 20 & 17 & 19 &\end{array} \nonumber \]

or in set notation as:

\[ \{18,18,19,19,19,18,22,20,18,18,17,19,18,24,18,20,18,21,20,17,19\} \nonumber \]

A data set can also be presented by means of a data frequency table, a table in which each distinct value \(x\) is listed in the first row and its frequency \(f\), which is the number of times the value \(x\) appears in the data set, is listed below it in the second row.

Example \(\PageIndex{2}\)

The data set of the previous example is represented by the data frequency table

\[\begin{array}{c|cccccc}x & 17 & 18 & 19 & 20 & 21 & 22 & 24 \\ \hline f & 2 & 8 & 5 & 3 & 1 & 1 & 1\end{array} \nonumber \]

The data frequency table is especially convenient when data sets are large and the number of distinct values is not too large.

Key Takeaway

• Data sets can be presented either by listing all the elements or by giving a table of values and frequencies.

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Data Presentation

After completing this chapter, you can understand the following:

• The meaning of data classification and its kinds.
• The different methods of condensing the data collected.
• Method of converting the collected data in the form of table.
• Various graphical representations of data and its importance.

3.1 INTRODUCTION

The successful use of data collected depends on the way in which it is arranged, displayed and summarized. As a part of it, this chapter is going to discuss the presentation and condensation of data.

3.2 CLASSIFICATION OF DATA

It is the process of arranging the data based on the similarities and dissimilarities. It is nothing but sorting.

3.2.1 Types of Classification

The data can be classified ...

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Presentation of Quantitative Data

• First Online: 01 January 2010

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• Hector Guerrero 2  

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We often think of data as being strictly numerical values, and in business, those values are often stated in terms of dollars. Although data in the form of dollars are ubiquitous, it is quite easy to imagine other numerical units: percentages, counts in categories, units of sales, etc. This chapter, and Chap. 3 , discusses how we can best use Excel’s graphics capabilities to effectively present quantitative data ( ratio and interval ), whether it is in dollars or some other quantitative measure, to inform and influence an audience. In Chaps. 4 and 5 we will acknowledge that not all data are numerical by focusing on qualitative ( categorical/nominal or ordinal ) data. The process of data gathering often produces a combination of data types, and throughout our discussions it will be impossible to ignore this fact: quantitative and qualitative data often occur together.

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Guerrero, H. (2010). Presentation of Quantitative Data. In: Excel Data Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10835-8_2

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CHAPTER 3 Data Description

Jul 25, 2014

1.17k likes | 1.37k Views

CHAPTER 3 Data Description. OUTLINE. 3-1 Introduction 3-2 Measures of Central Tendency 3-3 Measures of Variation 3-4 Measures of Position 3-5 Exploratory Data Analysis. OBJECTIVES. Summarize data using the measures of central tendency, such as the mean, median, mode, and midrange.

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• specific population
• population mean
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• exploratory data analysis

Presentation Transcript

CHAPTER 3Data Description

OUTLINE • 3-1 Introduction • 3-2 Measures of Central Tendency • 3-3 Measures of Variation • 3-4 Measures of Position • 3-5 Exploratory Data Analysis

OBJECTIVES • Summarize data using the measures of central tendency, such as the mean, median, mode, and midrange. • Describe data using the measures of variation, such as the range, variance, and standard deviation.

OBJECTIVES • Identify the position of a data value in a data set using various measures of position, such as percentiles, deciles and quartiles. • Use the techniques of exploratory data analysis, including stem and leaf plots, box plots, and five-number summaries to discover various aspects of data.

3-1 Introduction • A statisticis a characteristic or measure obtained by using the data values from a sample. • A parameteris a characteristic or measure obtained by using the data values from a specific population.

3-2 Measures of Central Tendency • Mean • Median • Mode • Mid-range

3-2 The mean (arithmetric average) • The mean is defined to be the sum of the data values divided by the total number of values.

3-2 The Sample Mean • The symbol X represents the sample mean. X is read as “X - bar”. The Greek symbol Σ is read as “sigma” and means “to sum”.

Example The following data represent the annual chocolate sales (in millions of RM) for a sample of seven states in Malaysia. Find the mean. RM2.0, 4.9, 6.5, 2.1, 5.1, 3.2, 16.6

3-2 The Population Mean • The Greek symbol µ represents the population mean. The symbol µ is read as “mu”. N is the size of the finite population.

Example A small company consists of the owner, the manager, the salesperson, and two technicians. Their salaries are listed as RM 50,000, 20,000, 12,000, 9,000 and 9,000 respectively. Assume this is the population, find the mean.

Question In a random sample of 7 ponds, the number of fishes were recorded as the following, find the mean. 23 56 45 36 28 33 37

3-2 The Sample Mean for an Ungrouped Frequency Distribution • The mean for an ungrouped frequency distribution is given by f = frequency of the corresponding value X n = f

Example The scores of 25 students on a 4-point quiz are given in the table. Find the mean score.

Question The number of balls in 17 bags were counted. Find the mean

3-2 The Sample Mean for a Grouped Frequency Distribution • The mean for grouped frequency distribution is given by Xm= class midpoint = (UCL + LCL) / 2

Example The lengths of 40 bean pods were showed in the table. Find the mean.

Question Time (in minutes) that needed by a group of students to complete a game are shown as below. Find the mean.

Most common errors:

Example Mean = (f•X) / n = 23 / 12 = 1.92 Mean = (f • X) / n = (12 x 10) / 12 = 10

3-2 The Median • When a data set is ordered, it is known as a data array. • The median is defined to be the midpoint of the data array. • The symbol used to denote the median is MD.

Example 1 The ages of seven preschool children are 1, 3, 4, 2, 3, 5, and 1. Find the median. 1. Arrange the data in order. 2. Select the middle point.

Data array: 1, 1, 2, 3, 3, 4, 5 Median The median (MD) age = 3 years.

In the previous example, there was an odd number of values in the data set. • In this case it is easy to select the middle number in the data array. • When there is an even number of values in the data set, the median is obtained by taking the average of the two middle numbers.

Example 2 Six customers purchased these numbers of magazines: 1, 7, 3, 2, 3, 4. Find the median. 1. Arrange the data in order. 2. Select the middle point.

Data array: 1, 2, 3, 3, 4, 7 Median • The median (MD) = 3 + 3 • 2 • = 3

3-2 The Median - Ungrouped Frequency Distribution • For an ungrouped frequency distribution, find the median by examining the cumulative frequencies to locate the middle value. • If n is the sample size, compute n/2. Locate the data point where n/2 values fall below and n/2 values fall above.

Example LRJ Appliance recorded the number of VCRs sold per week over a one-year period.

Solution • To locate the middle point, divide n by 2; 24/2 = 12. • Locate the point where 12 values would fall below and 12 values will fall above. • Consider the cumulative distribution. • The 12th and 13th values fall in class 2. Hence MD = 2.

This class contains the 5th through the 13th values. Median

3-2 The Median - Grouped Frequency Distribution • For grouped frequency distribution, find the median by using the formula as shown below: Median, MD = Lm + (W) n = sum of frequencies cf = cumulative frequency of the class immediately preceding the median class f = frequency of the median class w = class width of the median class Lm = Lower class boundary of the median class

Example Find the median by using the following data.

To locate the halfway point, divide n by 2; 17/2 = 8.5 ≈ 9. • Find the class that contains the 9th value. This will be the median class. • Consider the cumulative distribution. The median class will then be 26-30.

n = 17 cf = 8 f = 4 w = 30.5 – 25.5 = 5 = L 25.5 m - ( n 2 ) cf (17 / 2) – 8 = + + MD ( w ) L = ( 5 ) 25 . .5 f 4 m = 26.125.

Question Find the median by using the following data.

3-2 The Mode • The mode is defined to be the value that occurs most often in a data set. • A data set can have more than one mode. • A data set is said to have no mode if all values occur with equal frequency.

Example 1 Find the mode for the number of children per family for 10 selected families. Data set: 2, 3, 5, 2, 2, 1, 6, 4, 7, 3. Ordered set: 1, 2, 2, 2, 3, 3, 4, 5, 6, 7. Mode: 2.

Example 2 • Six strains of bacteria were tested to see how long they could remain alive outside their normal environment. The time, in minutes, is given below. Find the mode. • Data set: 2, 3, 5, 7, 8, 10. • There is no mode since each data value occurs equally with a frequency of one.

Example 3 • Eleven different automobiles were tested at a speed of 15 mph for stopping distances. The distance, in feet, is given below. Find the mode. • Data set: 15, 18, 18, 18, 20, 22, 24, 24, 24, 26, 26. • There aretwo modes (bimodal). The values are 18and 24.

3-2 The Mode – Ungrouped Frequency Distribution • Example Find the mode by using the following data. Highest frequency Mode

The mode for grouped data is the modal class. • The modal class is the class with the largest frequency.

3-2 The Mode – Grouped Frequency Distribution Example Find the mode by using the following data. Modal Class Highest frequency

3-2 The Midrange • The midrangeis found by adding the lowest and highest values in the data set and dividing by 2. • The midrange is a rough estimate of the middle value of the data. • The symbol that is used to represent the midrange isMR.

Example 1 • Last winter, the city of New York, reported the following number of water-line breaks per month. The data is as follows: 2, 3, 6, 8, 4, 1. Find the midrange. MR = (1 + 8)/2 = 4.5. • Note:Extreme values influence the midrange and thus may not be a typical description of the middle.

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Chapter 3 PRESENTATION, ANALYSIS, AND INTERPRETATION OF DATA

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Consider the accompanying contingency table, which presents the results of an advertising survey about the use of credit by Martan Oil Company customers. a. How many customers were surveyed? b. Why are these bivariate data? What type of variable is each one? c. How many customers preferred to use an oilcompany card? d. How many customers made 20 or more purchases last year? e. How many customers preferred to use an oilcompany card and made between five and nine purchases last year? f. What does the 80 in the fourth cell in the second row mean?

The June 2009 unemployment rates for eastern and western U.S. states were as follows:$$\begin{array}{lcccccc}\hline \text { Eastern } & 8.0 & 10.6 & 10.1 & 7.3 & 9.2 & 11.0 & 12.1 & 7.2 \\\text { Western } & 8.7 & 11.6 & 8.4 & 6.4 & 12.0 & 12.2 & 5.7 & 9.3 \\\hline\end{array}$$.Display these rates as two dotplots using the same scale; compare means and medians.

What effect does the minimum amount have on the interest rate being offered on 3 -month certificates of deposit (CDs)? The following are advertised rates of return, $y,$ for a minimum deposit of $\$ 500, \$ 1000$ $\$ 2500, \$ 5000,$ or $\$ 10,000, x .$ (Note that $x$ is in $\$ 100$ and $y$ is annual percentage rate of return.)$$\begin{array}{cc|cc|cc} \text { Min Deposit } & \text { Rate } & \text { Min Deposit } & \text { Rate } & \text { Min Deposit } & \text { Rate } \\\hline 100 & 0.95 & 25 & 1.00 & 25 & 0.75 \\100 & 1.24 & 50 & 1.00 & 10 & 0.75 \\10 & 1.24 & 100 & 1.00 & 100 & 0.70 \\10 & 1.15 & 5 & 1.00 & 5 & 0.64 \\100 & 1.10 & 10 & 1.00 & 10 & 0.50 \\50 & 1.09 & 10 & 0.80 & 100 & 0.35 \\100 & 1.07 & 10 & 0.75 & 25 & 0.35 \\ 5 & 1.00 & 10 & 0.75 & 5 & 0.99 \\25 & 0.75 & & & & \\\hline\end{array}$$.a. Prepare a dotplot of the five sets of data using a common scale. b. Prepare a 5 -number summary and a boxplot of the five sets of data. Use the same scale for the boxplots. c. Describe any differences you see among the three sets of data.

Can a woman's height be predicted from her mother's height? The heights of some mother-daughter pairs are listed; $x$ is the mother's height and $y$ is the daughter's height.$$\begin{array}{l|llllllllll}\hline x & 63 & 63 & 67 & 65 & 61 & 63 & 61 & 64 & 62 & 63 \\y & 63 & 65 & 65 & 65 & 64 & 64 & 63 & 62 & 63 & 64 \\\hline\end{array}$$,$$\begin{array}{l|lllllllllll} \hline \boldsymbol{x} & 64 & 63 & 64 & 64 & 63 & 67 & 61 & 65 & 64 & 65 & 66 \\ \boldsymbol{y} & 64 & 64 & 65 & 65 & 62 & 66 & 62 & 63 & 66 & 66 & 65 \\\hline\end{array}$$. a. Draw two dotplots using the same scale and showing the two sets of data side by side. b. What can you conclude from seeing the two sets of heights as separate sets in part a? Explain. c. Draw a scatter diagram of these data as ordered pairs. d. What can you conclude from seeing the data presented as ordered pairs? Explain.

The following tables list the ages, heights (in inches), and weights (in pounds) of the players on the 2009 roster for the National Hockey League teams Boston Bruins and Edmonton Oilers.a. Compare each of the three variables-height, weight, and age - using either a dotplot or a histogram (use the same scale). b. $\quad$ Based on what you see in the graphs in part a, can you detect a substantial difference between the two teams in regard to these three variables? Explain. c. Explain why the data, as used in part a, are not bivariate data.

Consider the two variables of a person's height and weight. Which variable, height or weight, would you use as the input variable when studying their relationship? Explain why.

Draw a coordinate axis and plot the points (0,6) $(3,5),(3,2),$ and (5,0) to form a scatter diagram. Describe the pattern that the data show in this display.

Does studying for an exam pay off? a. Draw a scatter diagram of the number of hours studied, $x,$ compared with the exam grade received, $y$.$$\begin{array}{l|rrrrr}\hline \boldsymbol{x} & 2 & 5 & 1 & 4 & 2 \\ \boldsymbol{y} & 80 & 80 & 70 & 90 & 60 \\\hline\end{array}$$.b. Explain what you can conclude based on the pattern of data shown on the scatter diagram drawn in part a. (Retain these solutions to use in Exercise $3.55,$ p. 157 )

Americans Love Their Automobiles" (Applied Example 3.4 on p. 128) to answer the following questions: a. Name the two variables used. b. Does the scatter diagram suggest a relationship between the two variables? Explain. c. What conclusion, if any, can you draw from the appearance of the scatter diagram?

Growth charts are commonly used by a child's pediatrician to monitor a child's growth. Consider the growth chart that follows.a. What are the two variables shown in the graph? b. What information does the ordered pair (3,87) represent?an two c. Describe how the pediatrician might use this chart and when what types of conclusions might be based on the information displayed by it.

Draw a scatter diagram showing height, $x,$ and weight, $y,$ for the Boston Bruins hockey team, using the data in Exercise 3.12 7d se b. Draw a scatter diagram showing height, $x,$ and $\Rightarrow ? \quad$ weight, $y,$ for the Edmonton Oilers hockey team using the data in Exercise 3.12 5), c. Explain why the data, as used in parts a and b, are n. bivariate data.

The accompanying data show the number of hours, $x,$ studied for an exam and the grade 50 received, $y(y \text { is measured in tens; that is, } y=8$ means that the grade, rounded to the nearest 10 points, is 80 ). Draw the scatter diagram. (Retain this solution to use in Exercise $3.37,$ p. $143 .$ )

An experimental psychologist asserts that the older a child is, the fewer irrelevant answers he or she will give during a controlled experiment. To investigate this claim, the following data were collected. Draw a scatter diagram. (Retain this solution to use in Exercise $3.38, p .143 .)$. $$\begin{array}{l|rrrrrrrrrr}\hline \text { Age, } x & 2 & 4 & 5 & 6 & 6 & 7 & 9 & 9 & 10 & 12 \\ \text { Irr Answers, } y & 12 & 13 & 9 & 7 & 12 & 8 & 6 & 9 & 7 & 5 \\\hline\end{array}$$.

A sample of 15 upper-class students whe commute to classes was selected at registration. They were asked to estimate the distance $(x)$ and the time $(y)=$ required to commute each day to class (see the following table).$$\begin{array}{cc|cc}\begin{array}{c}\text { Distance, } x \\\text { (nearest mile) }\end{array} & \begin{array}{c}\text { Time, } y \\\text { (nearest }5 \text { minutes }) \end{array} & \begin{array}{c}\text { Distance, } x \\\text { (nearest mile) }\end{array} & \begin{array}{c}\text { Time, } y \\\text { (nearest } 5 \text { minutes } \end{array} \\\hline 18 & 20 & 2 & 5 \\8 & 15 & 15 & 25 \\20 & 25 & 16 & 30 \\5 & 20 & 9 & 20 \\5 & 15 & 21 & 30 \\11 & 25 & 5 & 10 \\9 & 20 & 15 & 20 \\10 & 25 & & \\\hline \end{array}$$.a. Do you expect to find a linear relationship between the two variables commute distance and commute time? If so, explain what relationship you expect. b. Construct a scatter diagram depicting these data. c. Does the scatter diagram in part b reinforce what you expected in part a?

Refer to the 2009 4-wheel-drive, 6-cylinder SUVs chart in Applied Example 3.4 on page 128 and the two variables gas tank capacity, $x,$ and the cost to fill it, $y$ a. If you were to draw scatter diagrams of these two variables, on the same graph but separate, for the SUVs that use regular and premium gasoline, do you think the two sets of data would be distinguishable? Explain what you anticipate seeing. b. Construct a scatter diagram of tank capacity, $x,$ and fill-up cost, $y,$ for the SUVs using regular gasoline. c. Construct a scatter diagram of tank capacity, $x,$ and fill-up cost, $y,$ for the SUVs using premium gasoline on the scatter diagram for part b. d. Are the two sets of data distinguishable? e. How does your answer in part a compare to your answer in part d? Explain any difference.

Baseball stadiums vary in age, style and size, and many other ways. Fans might think of the size of a stadium in terms of the number of seats, while players might measure the size of a stadium in terms of the distance from home plate to the centerfield fence.$$\begin{array}{ll|cc|cc}\text { Seats } & \text { CF } & \text { Seats } & \text { CF } & \text { Seats } & \text { CF } \\\hline 38,805 & 420 & 36,331 & 434 & 40,950 & 435 \\41,118 & 400 & 43,405 & 405 & 38,496 & 400 \\56,000 & 400 & 48,911 & 400 & 41,900 & 400 \\45,030 & 400 & 50,449 & 415 & 42,271&404 \\34,077 & 400 & 50,091 & 400 & 43,647 & 401 \\40,793 & 400 & 43,772 & 404 & 42,600 & 396 \\56,144 & 408 & 49,033 & 407 & 46,200 & 400 \\50,516 & 400 & 47,447 & 405 & 41,222 & 403 \\40,615 & 400 & 40,120 & 422 & 52,355 & 408 \\48,190 & 406 & 41,503 & 404 & 45,000 & 408 \\\hline\end{array}$$.Is there a relationship between these two measurements of the "size" of the 30 Major League Baseball stadiums? a. What do you think you will find? Bigger fields have more seats? Smaller fields have more seats? No relationship between field size and number of seats? A strong relationship between field size and number of seats? Explain. b. Construct a scatter diagram. c. Describe what the scatter diagram tells you, including a reaction to your answer in part a.

Most adult Americans drive. But do you have any idea how many licensed drivers there are in each U.S. state? The table here lists the number of male and female drivers licensed in each of 15 randomly selected U.S. states during 2007 $$\begin{array}{lcccc} \hline \text { Male } & \text { Femole } & \text { Male } & \text { Female } \\ \hline 17.92 & 17.10 & 59.07 & 54.62 \\5.18 & 5.10 & 2.38 & 2.33 \\21.24 & 21.85 & 15.01 & 16.26 \\10.03 & 10.15 & 75.98 & 75.86 \\14.52 & 14.82 & 8.32 & 8.20 \\15.91 & 15.59 & 25.26 & 23.53 \\3.74 & 3.62 & 2.05 & 1.93 \\6.77 & 6.89 & & \\\hline\end{array}$$.a. Do you expect to find a linear (straight-line) relationship between number of male and number of female licensed drivers per state? How strong do you anticipate this relationship to be? Describe. b. Construct a scatter diagram using $x$ for the number of male drivers and $y$ for the number of female drivers. c. Compare the scatter diagram to your expectations in part a. How did you do? Explain. d. Are there data points that look like they are separate from the pattern created by the rest of ordered pairs? If they were removed from the dataset, would the results change? What caused these point(s) to be separate from the others, but yet still be part of the extended pattern? Explain. e. Use the dataset for all 51 states to construct a scatter diagram. Compare the pattern of the sample of 15 to the pattern shown by all $51 .$ Describe in detail. f. Did the sample provide enough information for you to understand the relationship between the two variables in this situation? Explain.

Ronald Fisher, an English statistician $(1890-1962),$ collected measurements for a sample of 150 irises. Of concern were five variables: species, petal width (PW), petal length (PL), sepal width (SW), and sepal length (SL) (all in $\mathrm{mm}$ ). Sepals are the outermost leaves that encase the flower before it opens. The goal of Fisher's experiment was to produce a simple function that could be used to classify flowers correctly. A random sample of his complete data set is given in the accompanying table.a. Construct a scatter diagram of petal length, $x,$ and petal width, $y .$ Use different symbols to represent the three species.. b. Construct a scatter diagram of sepal length, $x,$ and sepal width, $y .$ Use different symbols to represent the three species. c. Explain what the scatter diagrams in parts a and b portray.

Total solar eclipses actually take place nearly as often as total lunar eclipses, but the former are visible over a much narrower path. Both the path width and the duration vary substantially from one eclipse to the next. The table below shows the duration (in seconds) and path width (in miles) of 44 total solar eclipses measured in the past and those projected to the year 2010 :a. Draw a scatter diagram showing duration, $y,$ and path width, $x,$ for the total solar eclipses. b. How would you describe this diagram? c. The durations and path widths for the years $2006-2009$ were projections. The recorded values were:Compare the recorded values to the projections. Comment on accuracy. $$\begin{array}{lll} \text { Year } & \text { Path Width } & \text { Duration } \\ \hline 2006 & 65 \text { miles } & 247 \text { sec } \\ 2008 & 147 \text { miles } & 147 \text { sec } \\ 2009 & 160 \text { miles } & 399 \text { sec } \end{array}$$.Compare the recorded values to the projections. Comment on accuracy.