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soundofheaven.info Introduction to Business Statistics, Seventh. Edition. Ronald M. .. Also included, in pdf format, are Chapter 21, Ethics in Statistical Analysis. Introduction to Business Statistics, Seventh Edition Ronald M. Weiers Vice President of Editorial, Business: Jack W. Calhoun Publisher: Joe Sabatino Sr. Introduction-To-Business-Statistics-7Th-Edition Weiers Also included in pdf format are Chapter 21 Ethics in Statistical Analysis and Reporting.
There are a number of guidelines for constructing a frequency distribution. Brief Applied Calculus. When data values are in decimal form such as 3. One result is inevitable: Comment on whether any of the corporate sites seemed to require especially long or short times for the annual report to be found. As you might expect much of this text will be devoted to the concept and methods of inferential statistics. Introduction and Background The Pictogram Using symbols instead of a bar the pictogram can describe frequencies or other values of interest.
If possible the classes should have equal widths. Unequal class widths make it dif- ficult to interpret both frequency distributions and their graphical presen tations. Selecting the number of classes to use is a subjective process. If we have too few classes important characteristics of the data may be buried within the small number of categories. If there are too many classes many categories will contain either zero or a small number of values. In general about 5 to 15 classes will be suitable.
Whenever possible class widths should be round numbers e. For the highway speed data selecting a width of 2.
If possible avoid using open-end classes. These are classes with either no lower limit or no upper limit—e. Such classes may not always be avoidable however since some data may include just a few values that are either very high or very low compared to the others.
All values are at least 50 but less than The number of motorists with a speed in this category. The difference between the lower class limit and that of the next higher class or 55 minus The midpoint of the interval this can be calculated as the lower limit plus half the width of the interval or 50 1 0. Another useful approach to data expression is the relative frequency distribution which describes the proportion or percentage of data values that fall within each category.
The relative frequency distribution for the speed data is shown in Table 2. Relative frequencies can be useful in comparing two groups of unequal size since the actual frequencies would tend to be greater for each class within the larger group than for a class in the smaller one.
For example if a frequency distri- bution of incomes for physicians is compared with a frequency distribution for business executives more executives than physicians would be likely to fall into a given class. Relative frequency distributions would convert the groups to the same size: Relative frequencies will play an important role in our discussion of probabilities in Chapter 5. Cumulative Frequency Distribution. Another approach to the frequency dis- tribution is to list the number of observations that are within or below each of the classes.
This is known as a cumulative frequency distribution. When cumula- tive frequencies are divided by the total number of observations the result is a cumulative relative frequency distribution.
Examining this column we can readily see that Cumulative percentages can also operate in the other direction i. Based on Table 2. TABLE 2. Introduction and Background The Histogram The histogram describes a frequency distribution by using a series of adjacent rectangles each of which has a length proportional to either the frequency or the relative frequency of the class it represents. The histogram in part a of Fig- ure 2. The lower class limits e.
The tallest rectangle in part a of Figure 2. The relative heights of the rectangles visually demonstrate how the frequencies tend to drop off as we proceed from the 60—under 65 class to the 65—under 70 class and higher. The Frequency Polygon Closely related to the histogram the frequency polygon consists of line segments connecting the points formed by the intersections of the class marks with the class frequencies.
Relative frequencies or percentages may also be used in constructing the figure. Empty classes are included at each end so the curve will intersect the horizontal axis. For the speed-measurement data in Table 2. Histogram frequency polygon and percentage ogive for the speed-measurement data summarized in Table 2.
Visual Description of Data 21 Compared to the histogram the frequency polygon is more realistic in that the number of observations increases or decreases more gradually across the various classes.
The two endpoints make the diagram more complete by allowing the frequencies to taper off to zero at both ends. Related to the frequency polygon is the ogive a graphical display providing cumulative values for frequencies relative frequencies or percentages. We can use the computer to generate a histogram as well as the underlying frequency distribution on which the histogram is based. Computer Solutions 2. The name of the variable Speed is in cell A1 and the speeds are in the cells immediately below.
Type Bin into cell C1.
Enter the bin cutoffs 45 to 90 in multiples of 5 into C2: Alternatively you can skip this step if you want Excel to generate its default frequency distribution. From the Data ribbon click Data Analysis in the rightmost menu section.
Within Analysis Tools click Histogram. Click OK. Enter the data range A1: A into the Input Range box. If you entered the bin cutoffs as described in step 2 enter the bin range C1: C11 into the Bin Range box.
Click to place a check mark in the Labels box. This is because each variable has its name in the first cell of its block. Select Output Range and enter where the output is to begin—this will be cell E1. Click to place a check mark into the Chart Output box. Within the chart click on the word Bin. Click again and type in mph. Right-click on any one of the bars in the chart. Click Close. You can further improve the appearance by clicking on the chart and changing fonts item locations such as the key in the lower right or the background color of the display.
In the printout shown here we have also enlarged the display and moved it slightly to the left. Introduction and Background Excel and Minitab differ slightly in how they describe the classes in a frequency distribution. If you use the defaults in these programs the frequency distributions may differ slightly whenever a data point happens to have exactly the same value as one of the upper limits because 1.
This was not a problem with the speed data and Computer Solutions 2. The name of the variable Speed is in column C1 and the speeds are in the cells immediately below. Click Graph. Click Histogram. Select Simple and click OK. Within the Histogram menu indicate the column to be graphed by entering C1 into the Graph variables box. Click the Labels button then click the Data Labels tab. Select Use y-value labels. This will show the counts at the top of the bars.
On the graph that appears double-click on any one of the numbers on the horizontal axis. Click the Binning tab. In the Interval Type submenu select Cutpoint. This provides intervals from 45 to 90 with the width of each interval being 5.
Visual Description of Data 23 2. Age Years Licensed Drivers Millions under 20 The New York Times Almanac p. Age Years Number of Deaths Thousands under 15 Federal Deposit Insurance Corporation has listed their total deposits billions of dollars as follows.
Deposits Deposits Deposits AL Each of the techniques can be carried out either manually or with the computer and statistical software. The Stem-and-Leaf Display The stem-and-leaf display a variant of the frequency distribution uses a subset of the original digits as class descriptors. The technique is best explained through 2. The World Almanac and Book of Facts p. Department of Agri culture the distribution of U. Farms Annual Sales Thousands under —under —under —under —under —under 82 —under 42 or more 29 Convert this information to a a.
Relative frequency distribution. Exercises 2. Use your computer statistical software to generate a frequency distribution and histogram describ- ing this information. Do any of the portfolio values seem to be especially large or small compared to the others 2. In randomly selecting and visiting 80 corporate websites an investor needed an average of The 80 times are in data file XR Generate a frequency distribution and histogram describing this information. Comment on whether any of the corporate sites seemed to require especially long or short times for the annual report to be found.
Assume that an executive wishing to replicate this study within her own corporation directs information technology personnel to find out the number of e-mails each of a sample of office workers received yesterday with the results as provided in the data file XR Generate a frequency distribution and histogram describing this information and comment on the extent to which some workers appeared to be receiving an especially high or low number of e-mails.
Visual Description of Data 25 a few examples. The raw data are the numbers of Congressional bills vetoed during the administrations of seven U. By using the digits in the data values we have identified five different categories 30s 40s 50s 60s and 70s and can see that there are three data values in the 30s two in the 40s one in the 60s and one in the 70s.
Like the frequency distribution the stem-and-leaf display allows us to quickly see how the data are arranged. For example none of these presidents vetoed more than 44 bills with the exceptions of Presidents Ford and Reagan who vetoed 66 and 78 respectively.
Compared with the frequency distribution the stem-and-leaf display provides more detail since it can describe the individual data values as well as show how many are in each group or stem. When the stem-and-leaf display is computer generated the result may vary slightly from the previous example of the presidential vetoes. For example Minitab may do the following: Break each stem into two or more lines each of which will have leaves in a given range.
In Computer Solutions 2. The first line includes those with a leaf that is 0 to 4 i. Include stem-and-leaf figures for outliers. An outlier is a data value very dis- tant from most of the others. The stem-and-leaf display shows just two figures for each data value.
Introduction and Background 2 all of the values must be expressed in terms of the same stem digit and the same leaf digit. If we were to deviate from either of these rules the resulting dis- play would be meaningless. When data values are in decimal form such as 3. For example the number 3. The location of the decimal point would have to be considered during interpretation of the display.
Click on A1 and drag to A to select the label and data values in cells A1: Ensure that the Input Range box con- tains the range you specified in step 1. Click Labels. The variable Speed is in column C1. Click Stem-And-Leaf. Enter C1 into the Variables box. Visual Description of Data 27 The Dotplot The dotplot displays each data value as a dot and allows us to readily see the shape of the distribution as well as the high and low values.
Click Dotplot. Select One Y Simple. We will also provide several Computer Solutions to guide you in using Excel and Minitab to generate some of the more common graphical presenta- tions. These are just some of the more popular approaches. There are many other possibilities. The Bar Chart Like the histogram the bar chart represents frequencies according to the relative lengths of a set of rectangles but it differs in two respects from the histogram: What is the original set of data 2z 3z 4z 5z47 2.
Is it possible to determine the exact values in the original data from this display If so list the data values. If not provide a set of data that could have led to this display. The appraisals in thousands of dollars are contained in data file XR Construct stem-and-leaf and dotplot displays describing the appraisal data. Construct stem-and-leaf and dotplot displays for these data. Visual Description of Data 29 The Line Graph The line graph is capable of simultaneously showing values of two quantitative variables y or vertical axis and x or horizontal axis it consists of linear segments connecting points observed or measured for each variable.
When x represents time the result is a time series view of the y variable. Even more information can be presented if two or more y variables are graphed together. The labels Certificate Type and Number thousands have already been entered into A3 and C3 respectively.
The name for the first category has already been entered into A4 and its corresponding numerical value thousands of pilots has been entered into C4. We have continued downward entering category names into column A and corresponding numerical values into column C until all category names and their values have been entered into columns A and C.
Click on A4 and drag to C7 to select cells A4: From the Insert ribbon click Bar from the Charts submenu. From the next menu click the first choice Clustered Bar from the 2-D Bar section. To further improve the appearance click on the chart and drag the borders to expand it vertically and horizon- tally. Additional editing can include insertion of the desired labels: Double-click within the chart area. The labels Certificate Type and Number thousands have already been entered at the top of columns C1 and C2 respectively.
The names of the categories have already been entered into C1 and the numerical values thousands of pilots have been entered into C2. Click Bar Chart. In the Bars represent box select Values from a table. Select One column of values Simple. Within the Graph variables menu enter C2. Enter C1 into the Categorical variable box.
Click Scale. Select the Axes and Ticks tab and select Transpose value and category scales. Visual Description of Data 31 The Pie Chart The pie chart is a circular display divided into sections based on either the number of observations within or the relative values of the segments.
If the pie chart is not com- puter generated it can be constructed by using the principle that a circle contains degrees. The angle used for each piece of the pie can be calculated as follows: Number of degrees for the category 5 Relative value of the category 3 For example if 25 of the observations fall into a group they would be represented by a section of the circle that includes 0.
The labels and data are already entered as shown in the printout. Click on A4 and drag to B9 to select cells A4: Within the two columns the variable to be represented by the vertical axis should always be in the column at the right. From the Insert ribbon click Scatter from the Charts menu. From the Scatter submenu click the fifth option Scatter with Straight Lines. Further appearance improvements can be made by clicking on the chart and dragging the borders to expand it vertically and horizontally.
The labels Year and Net Income have already been entered at the top of columns C1 and C2 respectively. The years and the numerical values have been entered into C1 and C2 respectively.
Click Scatterplot. Select With Connect Line. Enter C2 into the first line of the Y vari- ables box. Enter C1 into the first line of the X variables box. Double-click on the chart title and the vertical-axis labels and revise as shown. Home Depot Corporation Annual Report pp. Introduction and Background The Pictogram Using symbols instead of a bar the pictogram can describe frequencies or other values of interest.
Figure 2. In the diagram each truck represents about When setting up a pictogram the choice of symbols is up to you. This is an important consid- eration because the right or wrong symbols can lend nonverbal or emotional content to the display. This chart shows how soft drink sales millions of cases in Central America increased from through Panamco Annual Report p.
Total Soft Drink Sales in millions of unit cases Visual Description of Data 33 1. The segment names have already been entered as shown in the display as have the sales for each segment. Click on A4 and drag to B7 to select cells A4: From the Insert ribbon and its Charts menu click Pie. Click on the first option Pie in the 2-D Pie menu. Additional editing can include insertion of the desired chart title: From the Layout ribbon use the Chart Title selection within the Labels menu.
The labels Segment and Sales have already been entered at the top of columns C1 and C2 respectively. The names of the segments have already been entered into C1 and the numerical values sales billions of dollars have been entered into C2. Click Pie Chart. Select Chart values from a table. Enter C2 into the Summary variables box. Double-click on the chart title and revise as shown. The Sketch Varying in size depending on the frequency or other numerical value displayed the sketch is a drawing or pictorial representation of some symbol relevant to the data.
This approach will be demonstrated in part c of Figure 2. Other Visuals The preceding approaches are but a few of the many possibilities for the visual description of information. One of these displays is shown in Figure 2. Introduction and Background The Abuse of Visual Displays Remember that visuals can be designed to be either emotionally charged or purposely misleading to the unwary viewer.
This capacity to mislead is shared by a great many statistical tests and descriptions as well as visual displays. We will consider just a few of the many possible examples where graphical methods could be viewed as misleading. Copyright Reprinted with permission. Part a shows the effect of compressing the data by using a high endpoint for the vertical axis.
In part b the change is exaggerated by taking a slice from the vertical axis. In part c although the sketch representing is Visual Description of Data 35 During Another strategy for achieving the same effect is to begin the vertical axis with a value other than zero.
In part c of Figure 2. Although the sketch on the right is This is because area for each sketch is height times width and both the height and the width are Because both height and width are increased the sketch for shipments has an area Over the years shown sales increased by only 1.
However in part b the starting 5 Data source: Introduction and Background Note: These graphs and charts can be done by hand but if possible use the computer and your statistical software.
World Almanac and Book of Facts p.
Country or Region of Manufacture market share North America Time Almanac p. Both lines will be plotted on the same graph.
Camden N. Construct a pie chart to summarize these contributions. Construct a bar chart to summarize these contributions. Why is it appropriate to construct a bar chart for these data instead of a histogram 2. Construct two separate pie charts to summarize telephone ownership and cable television service.
Using appropriate symbols that would reflect favor- ably on such an increase construct a pictogram to compare with Using appropriate symbols that would reflect unfa- vorably on such an increase construct a pictogram to compare with As a result it looks as though the company has done very well over the years shown. To examine whether a relationship exists we can begin with a graphical device known as the scatter diagram or scatterplot. Think of the scatter diagram as a sort of two-dimensional dotplot.
Each point in the diagram represents a pair of known or observed values of two variables generally referred to as y and x with y represented along the vertical axis and x represented along the horizontal axis. The two variables are referred to as the dependent y and independent x variables since a typical purpose for this type of analysis is to estimate or predict what y will be for a given value of x. A direct positive linear relationship between the variables as shown in part a of Figure 2.
The best-fit line is linear and has a positive slope with both y and x increasing together. An inverse negative linear relationship between the variables as shown in part b of Figure 2.
The best-fit line is linear and has a negative slope with y decreasing as x increases. In part d there is no relationship at all between the variables.
Introduction and Background 3. A curvilinear relationship between the variables as shown in part c of Figure 2. The best-fit line is a curve. As with a linear relationship a curvilinear relationship can be either direct positive or inverse negative. No relationship between the variables as shown in part d of Figure 2. The best-fit line is horizontal with a slope of zero and when we view the scatter dia- gram knowing the value of x is of no help whatsoever in predicting the value of y.
In this chapter we will consider only linear relationships between variables and there will be two possibilities for fitting a straight line i. The second more accurate approach is to use the computer and your statistical software to fit a straight line that is mathe matically optimum. A partial listing of the data for the 30 teams is shown here: San Diego Padres The two vari- ables do appear to be related—teams with a higher payroll did tend to win more games during the season.
It makes sense that better players will win more games and that better players must be paid more money thus it comes as no surprise that the higher-paid teams won more games. However the scatter diagram has provided us with a visual picture that reinforces our intuitive feelings about wins and dollars. Visual Description of Data 39 We can use Excel or Minitab to easily generate both a scatter diagram for the data and the linear equation that best fits the 30 data points.
The procedures and results are shown in Computer Solutions 2. Given a value for x we can use the equation to estimate a value for y. For example if a team had a payroll of 50 million we estimate that this team would have had This is obtained simply by substituting x 5 50 million into the equation and calculating an estimated value for y 5 wins.
We can interpret the slope of the equation—in this case 0. Accordingly an extra 10 million would tend to produce an extra 1. We can interpret the slope in terms of the type of linear relationship between the variables.
For the baseball data the slope is positive We can identify data points where the actual value of y is quite different from the value of y predicted by the equation. In this situation we might be interested in teams that overperformed won more games than their payroll would have predicted or underperformed won fewer games than their payroll would have predicted.
For example the Oakland Athletics played better than they were paid. Given their payroll of Teams with a higher payroll tended to win more games. Data sources: Click on B1 and drag to C31 to select cells B1: As with the line chart of the two columns the variable to be repre- sented by the vertical axis should always be in the column to the right.
From the Scatter submenu click the first option Scatter with only Markers. This optional step adds the best-fit straight line. Right-click on any one of the points in the scatter diagram. When the menu appears click Add Trendline. Along with the equation the display includes information we will be covering later in the text.
The labels Payroll and Wins have already been entered at the top of columns C2 and C3 respectively. The payroll values and the win totals have been entered into C2 and C3 respectively.
Click Stat. Select Regression. Click Fitted Line Plot. Enter C3 into the Response Y box. Enter C2 into the Predictor X box. Click to select the Linear model. Visual Description of Data 41 The nature of this chapter has allowed us to only briefly introduce the scatter diagram the best-fit linear equation and interpretation of the results.
Later in the text in Chapter 15 Simple Linear Regression and Correlation we will cover this topic in much greater detail. Click Options. Enter Wins vs. Payroll into the Title box. PC World July p. Draw a scatter diagram representing these data. Does there appear to be any relationship between the variables If so is the relationship direct or inverse 2. Introduction and Background a.
The banks the bailout they received and the additional cushion they were deemed to need are as shown below. The data are in file XR Generate a scatter diagram that includes the best-fit linear equation for these data.
Does there appear to be any relationship between the variables If so is the relationship direct or inverse c. Interpret the slope of the equation generated in part a. Viewing the scatter diagram and equation in part a do you think there are any employees who appear to own an unusually high or low amount of stock com- pared to the ownership that the equation would have predicted 2. Considering square footage as the independent variable and monthly rental fee as the dependent variable: If each person or item is also described by a quantitative variable e.
Visual Description of Data 43 EXAMPLE Tabulation Methods For 50 persons observed using a local automated teller machine ATM researchers have described the customers according to age category and gender category and have used a stopwatch to measure how long the customer required to complete his or her transactions at the machine.
In this situation age category and gender are considered to be nominal-scale category variables and time measured in seconds is a quantitative variable. The raw data are shown in part A of Table 2. An alternative would have been to express the counts in terms of percentages—e. Cross-Tabulation Contingency Table The cross-tabulation also known as the crosstab or the contingency table shows how many people or items are in combinations of categories.
The cross-tabulation in part C of Table 2. Because there are two category variables involved part C of Table 2. Cross-tabulations help us to identify and examine possible relationships between the variables. For example the crosstab in part C of Table 2.
Given these results bank management might want to interview customers to learn whether women in the middle and older age categories might have security concerns about the ATM location lighting or layout. In a useful application of the concept of cross-tabulation we can generate a tabular display that describes how a selected quantitative variable tends to differ from one category to another or from one combination of categories to another.
For example part D of Table 2. The average time for all 50 persons was From a gender perspective the average time for males Age cat- egory and gender are nominal- scale category variables and time is a quantitative variable measured in seconds. The cod- ing for the age and gender cat- egories follows: AgeCat 5 1 for age 30 2 for age 30—60 3 for age. Age Category: The variables are age category 1 5 30 2 5 30—60 and 3 5. Click on A1 and drag to C51 to select cells A1: From the Insert ribbon and its Tables menu click PivotTable.
In the Create PivotTable menu click Select a table or range. The cells A1: Click to select Existing Worksheet and enter D1 into the box. Click the AgeCat label at the right and drag it into the Row Labels rectangle. Click the Gender label at the right and drag it into the Column Labels rectangle. Click the Gender label again and drag it into the Values rectangle.
Right-click on any one of the data values within the table then select Value Field Settings. In the Summarize value field by box select Count. Click on the Row Labels cell and edit by entering AgeCat. Click on the Column labels cell and edit by entering Gender. Select Tables. Click Cross Tabulation and Chi-Square.
In the Display portion click to place a check mark next to Counts. Click the Seconds label at the right and drag it into the Values rectangle. In the Summarize value field by box select Average. Click on E3 and drag to G6 to select cells E3: Right-click within the field and select Number Format. Select Number and specify 2 in the Decimal places box. Click on the Column Labels cell and edit by entering Gender. Click on any cell within the table. From the Insert ribbon select Column from the Charts menu.
Select the first option Clustered Column from the 2-D Column choices. The chart can now be edited and the desired labels inserted: As in Computer Solutions 2. The corresponding age categories 1 5 30 2 5 30—60 and 3 5.
Click Descriptive Statistics. Visual Description of Data 47 3. Select Means. These characteristics along with the miles per gallon mpg achieved by each vehicle during the past month are listed here.
The data are also provided in file XR Construct a simple tabulation in which the counts are according to the type of engine.
Construct a cross-tabulation describing the fleet using type of engine and type of transmission as the categorization variables. Construct a display showing the average mpg accord- ing to type of engine and type of transmission. Introduction and Background 2. Construct a cross-tabulation describing the fleet using type of engine and whether the vehicle has air conditioning as the categorization variables. Construct a display showing the average mpg accord- ing to type of engine and whether the vehicle has air conditioning.
Do the categorization variables seem to be related to mpg If so how 2. Construct a simple tabulation in which the counts are according to the type of trans mission. Construct a cross-tabulation describing the fleet using type of transmission and whether the vehicle has air conditioning as the categorization variables. Construct a display showing the average mpg accord- ing to type of transmission and whether the vehicle has air conditioning.
Construct a simple tabulation in which the counts are according to the highest degree level offered.
Construct a cross-tabulation describing the schools using the highest degree level offered and whether the school is public or private as the categorization variables. Construct a display showing the average value for tuition and fees according to highest degree level offered and whether the school is public or private. Do the categorization variables seem to be related to the level of tuition and fees If so how 2.
Construct a simple tabulation in which the counts are according to the type of campus setting. Construct a cross-tabulation describing the schools using the type of campus setting and whether the school is public or private as the categorization variables. Construct a display showing the average value for tuition and fees according to type of campus setting and whether the school is public or private.
We have also included one more variable for each city: The New York Times Almanac pp. Construct a simple tabulation in which the counts are according to the grade on financial management.
Construct a cross-tabulation describing the cities using grade on financial management and grade on information technology as the categorization variables. Construct a display showing the average population size according to grade on financial management and grade on information technology. Do the categoriza- tion variables seem to be related to the level of popu- lation If so how 2. Construct a simple tabulation in which the counts are according to the grade on personnel policies.
Construct a cross-tabulation describing the cities using the grade on personnel policies and the grade on managing for results as the categorization variables. Construct a display showing the average population according to grade on personnel policies and the grade on managing for results. When data are quantitative they can be transformed to a frequency distribution or a histogram describing the number of observations occurring in each category.
Visual Description of Data 49 The set of classes in the frequency distribution must include all possible val- ues and should be selected so that any given value falls into just one category. In general between 5 and 15 classes are employed. A frequency distribution may be converted to show either relative or cumulative frequencies for the data.
In the dotplot values for a variable are shown as dots appearing along a single dimension. Frequency polygons ogives bar charts line graphs pie charts pictograms and sketches are among the more popular methods of visually summarizing data. As with many statistical methods the possibility exists for the purposeful distortion of graphical information. These are typically referred to as the dependent variable y and the independent variable x.
This type of anal ysis is carried out to fit an equation to the data to estimate or predict the value of y for a given value of x. These tabular methods can be extended to include the mean or other measures of a selected quantitative variable for persons or items within a category or combination of categories. Population Number of Cities —under —under —under —under —under 37 —under 25 or more 9 a.
How many cities have a population of at least but less than b.
How many cities have a population less than c. How many cities have a population of at least but less than What percentage of cities are in this group d. What is the class mark for the —under class e. Convert the table to a relative frequency distribution. These data are also provided in the f le XR Construct a stem-and-leaf display for these data. Construct a frequency distribution. Determine the interval width and the class mark for each of the classes in your frequency distribution.
Based on the frequency distribution obtained in part b draw a histogram and a relative frequency poly- gon to describe the data. Based on the frequency distribution obtained in part b draw a histogram and a relative frequency polygon. Percent Percent Percent AL 4.
Construct a frequency distribution for these data. Sales Thousands of Homes Total United Western Price of Home States States under 22 —under 54 —under 64 or over 94 41 Convert these data to relative frequency distributions one for the total United States the other for the western states. Do the results appear to suggest any differences between total U. The data consist of two-digit integers. From this display is it possible to determine the exact values in the original data If so list the data values.
Interpret the numbers in the leftmost column of the output. The data consist of three-digit integers. From this display is it possible to determine the exact val- ues in the original data If so list the data values.
The data are also in the f le XR Some would suggest that increases in advertis- ing should be accompanied by increases in sales.
Does your line graph support this 2. Use the same scale for each dotplot then comment on whether unemployment appears to have changed in terms of its range highest minus lowest or in the general level of unemployment perhaps a slight shift to the right or to the left from to Data for both years are in the f le XR News World Report provided two reputation scores with maximum 5 5.
The data values are listed in f le XR Construct a scatter diagram where the variables are the two kinds of reputation scores. Fit a linear equation to the scatter diagram. Is the slope positive or is it negative Is the sign of the slope consis- tent with your intuitive observation in part a 2. The percentages are listed in data f le XR Construct a scatter diagram using the household sav- ing rates in the United States and Canada as the two variables.
In the years when U. Construct a scatter diagram using the household sav- ing rates in the United States and Germany as the two variables.
Is the slope positive or is it negative Is the sign of the slope consis- tent with your intuitive observation in part a. Visual Description of Data 53 2. Following production f nished compressor units are tested to measure how much pressure they can exert pounds per square inch or psi. During the past few months compressors have been tested before ship- ment to customers and the resulting data are listed in f le XR Construct a simple tabulation in which the counts are according to which company supplied the mechanical components.
Based on the crosstab and means in part c would it seem that any of the f ve suppliers should be examined further with regard to the effect their product might have on the f nal pressure capabilities of the f nished product Explain your answer.
Based on the crosstab and means in part c would it seem that the two technicians might not be equally adept at the f nal assembly task Explain your answer. Luke Thorndike founder and current president of Thorn- dike Sports Equip ment had guided the business through 34 successful years and was now interested in bringing his favorite grandson Ted into the company.
You always were a high-strung kid. Thought you might like to join our tennis racquet division. She seems trustwor- thy but we might need to have some numbers to back up our claim if we decide to come out with the product. Thorndike has proposed Ted accepts the offer. He decides to mix 25 of the new balls with 25 of the old type have a golf pro hit all 50 of them at a driving range then measure how far each goes.
Introduction and Background Source: Data are based on actual responses obtained to this subset of the questions included in the survey town and mall identities have been disguised.
Using yard intervals beginning with Using the same intervals as in part 1 construct a fre- quency distribution for the distances traveled by the conventional ball. Place the frequency distribution for the new ball next to the one for the conventional ball.
A telephone survey has been conducted to identify strengths and weaknesses of these areas and to f nd out how they f t into the shopping activities of local residents. The respondents were also asked to provide information about themselves and their shopping habits. The variables in the survey were as follows: Springdale Mall Downtown West Mall or more 1 1 1 —under 2 2 2 —under 3 3 3 50—under 4 4 4 25—under 50 5 5 5 15—under 25 6 6 6 less than 15 7 7 7.
Visual Description of Data 55 C. High quality of goods 1 2 3 4 Low prices 1 2 3 4 Convenient shopping hours 1 2 3 4 Clean stores and surroundings 1 2 3 4 A lot of bargain sales 1 2 3 4 E.
High quality of goods 1 2 3 4 5 6 7 Convenient shopping hours 1 2 3 4 5 6 7 Clean stores and surroundings 1 2 3 4 5 6 7 Information about the Respondent Variables 26—30 Number of years of school completed: Marital status: Number of people in household: Introduction and Background Each respondent in this database is described by 30 variables.
As an example of their interpretation con- sider row number 1. This corresponds to respondent number 1 and contains the following information. Variable number 6 5 7. The respondent usually spends less than 15 when she shops at West Mall. Variable number 26 5 2. The respondent is a female. Variable number 19 5 7. Variable number 1 5 5. In applying some of the techniques from this chapter the following questions could provide insights into the perceptions and behavior of Springdale residents regarding the three shopping areas.
Do people tend to spend differently at the three areas Construct and compare frequency distributions for variables 4 5 and 6. To f nd out more about specif c strengths and weak- nesses of the areas set up a frequency distribution for variable 10 i.
Repeat this for variables 11—17 and interpret the results. Generate a cross-tabulation in which the categoriza- tion variables are variable 26 gender and variable 28 marital status. For each of the four subcategories in the cross-tabulation have your statistical software include the average for variable 30 respondent age. Interpret the results. Workers in the public sector used an average of Does the British private sector attract younger and healthier workers or is it possible that public-sector workers simply take advantage of the greater num- ber of sick days they are generally allowed According to a survey of U.
Human resource direc- tors suspect that only 28 of paid sick leave is actually used because of illness but they have no way of knowing for sure which absent employees are really sick. However the number of those who are actu- ally feeling under the weather may not necessarily match up with unscheduled absences. One in four workers report they feel sick days are equivalent to taking extra vacation days and treat them as such.
From this perspective union hourly workers use a median of 6. Try not to be absent. We will now examine statistical methods for describing typical values in the data as well as the extent to which the data are spread out. Introduced in Chapter 1 these descriptors are known as measures of central tendency and mea sures of dispersion: Chances are the typical subscriber to The Wall Street Journal earns more than the typical subscriber to Mad magazine.
By using measures of central tendency we can numerically describe the typical income of members of each group. The primary measures of central tendency dis- cussed in this chapter are the arithmetic mean weighted mean median and mode. Measures of dispersion allow us to numerically describe the scatter or spread of mea surements.
Among the measures of dispersion discussed in this chapter are the range quantiles mean absolute deviation variance and standard deviation. Chapter 2 also introduced the scatter diagram and best-fit linear equation as a graphical method of examining the possible linear relationship between two quantitative variables.
In this chapter we will extend this to two statistical mea- sures of association: As with the scatter diagram and best-fit equation these will be discussed in greater detail in Chapter 15 Simple Linear Regression and Correlation.
Chapter 3: This numerical descriptor has the purpose of describing the typical observation contained within the data. The Arithmetic Mean Defined as the sum of the data values divided by the number of observations the arithmetic mean is one of the most common measures of central tendency. The population mean applies when our data represent all of the items within the population. Population mean: Sample mean: As an example of the calculation of a population mean consider the following data for shipments of peanuts from a hypothetical U.
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