Gen Combo Ll Applied Statistics In Business & Economics; Connect Access Card
Gen Combo Ll Applied Statistics In Business & Economics; Connect Access Card
6th Edition
ISBN: 9781260260632
Author: David Doane, Lori Seward Senior Instructor of Operations Management
Publisher: McGraw-Hill Education
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Chapter 3, Problem 30CE

Below are batting averages of the New York Yankees players who were at bat five times or more in 2006. (a) Construct a frequency distribution. Explain how you chose the number of bins and the bin limits. (b) Make a histogram and describe its appearance. (c) Repeat, using a different number of bins and different bin limits. (d) Did your visual impression of the data change when you changed the number of bins? Explain.

Batting Averages for the 2006 New York Yankees

Chapter 3, Problem 30CE, Below are batting averages of the New York Yankees players who were at bat five times or more in

a.

Expert Solution
Check Mark
To determine

Construct a frequency distribution and explain how the number of bins and the bin limits are chosen.

Answer to Problem 30CE

Thefrequency distribution is tabulated below,

Bin limitsMid pointWidth

Frequency

f

PercentCumulative
LowerUpperFrequencyPercent
0< 0.070.0350.071414
0.07< 0.140.1050.070014
0.14< 0.210.1750.07416520
0.21< 0.280.2450.079361456
0.28< 0.350.3150.0710402496
0.35< 0.420.3850.071425100
Total      

Explanation of Solution

Calculation:

The given information is that, the data representsthebatting averages of the New York Yankees players who were at bat five times or more in 2006.

Frequency distribution:

It is a tabulation of n data values which are divided into k classes called bins. The bin limits are the cutoff points which defines each bin. These generally have equal interval and the limits do not overlap.

Step-by-step procedure to construct frequency distribution table is as follows:

  • • The smallest and largest data values are 0 and 0.417.
  • • Here the sample size is 25. By Sturge’s Rule, k=1+3.3log(n)

    k=1+3.3log(25)=1+4.6132=5.616

  • • Bin width is obtained by dividing the range by the number of bins.

    Binwidth=xmaxxmink=0.41706=0.06950.07

Hence, the bin width is 0.07.

  • • The minimum value in the data is 0 hence the first bin should start at 0.

Tally mark:

  • • Make a tally mark for each score in the corresponding class and continue for all reading times in the data.
  • • The number of tally marks in each class represents the frequency, f of that class.

Thus, the frequency distribution table is as follows:

Bin limitsTally

Frequency

f

Percent
LowerUpper
0< 0.07|1125×100=4
0.07< 0.14 0025×100=0
0.14< 0.21||||4425×100=16
0.21< 0.28||||||||9925×100=36
0.28< 0.35||||||||101025×100=40
0.35< 0.42|1125×100=4
Total 25 

Mid point:

The midpoint is the average of the lower limit and upper limit of a particular class. It is also called as class mark.

Midpoint=(Lowerclass limit+Upperclasslimit)2

Thus, the mid points for each class is tabulated below:

Bin limits

Frequency

f

Mid point
LowerUpper
0< 0.0710+0.072=0.035
0.07< 0.1400.07+0.142=0.105
0.14< 0.2140.14+0.212=0.175
0.21< 0.2890.21+0.282=0.245
0.28< 0.35100.28+0.352=0.315
0.35< 0.4210.42+0.492=0.455
Total25 

Cumulative frequency:

Cumulative frequency is the sum of all frequencies up to that class. The last class’s cumulative frequency is equal to the sample size n.

Thus, the cumulative frequency for each calss is tabulated below:

Bin limits

Frequency

f

Cumulative

frequency

LowerUpper
0< 0.0711
0.07< 0.1401+0=1
0.14< 0.2141+4=5
0.21< 0.2895+9=14
0.28< 0.351014+10=24
0.35< 0.42124+1=25
Total25 

Cumulative Relative frequency:

Bin limits

Cumulative

frequency

Cumulative

percent

LowerUpper
0< 0.071125×100=4
0.07< 0.141125×100=4
0.14< 0.215525×100=20
0.21< 0.28141425×100=56
0.28< 0.35242425×100=96
0.35< 0.42252525×100=100
Total  

b.

Expert Solution
Check Mark
To determine

Constuct a histogram and describe its appearance.

Answer to Problem 30CE

The histogram for the data is as follows,

Gen Combo Ll Applied Statistics In Business & Economics; Connect Access Card, Chapter 3, Problem 30CE , additional homework tip  1

Explanation of Solution

Calculation:

Software procedure:

Step-by-step procedure to obtain histogram using MINITAB software is as follows,

  • • Choose Graph > Histogram.
  • • Choose Simple, and then click OK.
  • • In Graph variables, enter the corresponding column of Yankees.
  • • Click Scale>Y-Scale Type > Percent
  • • Click OK.
  • • To modify the interval settings, double click on the horizontal axis of the graph. Then, select Binning>Cutpoint>Cutpoint Positions, in this box, enter the values for the cut points of the bin intervals (0, 0.07, 0.14, 0.21, 0.28, 0.35 and 0.42).

From the histogram it is observed that the tail on the left is more elongated than the tail on the right. Hence the distribution of the histogram seems to be skewed left.

c.

Expert Solution
Check Mark
To determine

Repeat the procedure using different number of bins and different bin limits.

Answer to Problem 30CE

Thefrequency distribution using 5 bins is as follows,

Bin limitsMid pointWidth

Frequency

f

PercentCumulative
LowerUpperFrequencyPercent
0< 0.10.050.11414
0.1< 0.20.150.1312416
0.2< 0.30.250.115601976
0.3< 0.40.350.15202496
0.4< 0.50.450.11425100
Total      

The histogram for 5 bins with 0.1 width is as follows,

Gen Combo Ll Applied Statistics In Business & Economics; Connect Access Card, Chapter 3, Problem 30CE , additional homework tip  2

Thefrequency distribution using 0.15 width and 9 bins is as follows,

Bin limitsMid pointWidth

Frequency

f

PercentCumulative
LowerUpperFrequencyPercent
0< 0.050.0250.051414
0.05< 0.100.0750.050014
0.10< 0.150.1250.051428
0.15< 0.200.1750.0528416
0.20< 0.250.2250.056241040
0.25< 0.300.2750.059361976
0.30< 0.350.3250.055202496
0.35< 0.400.3750.05002496
0.45< 0.450.4250.051425100
Total      

The histogram for 9 bins with width 0.05 is as follows,

Gen Combo Ll Applied Statistics In Business & Economics; Connect Access Card, Chapter 3, Problem 30CE , additional homework tip  3

Explanation of Solution

Calculation:

For 5 bins:

Here the number of bins are considered as 5 with width 0.1.

Bin limitsTally

Frequency

f

Percent
LowerUpper
0< 0.1|1125×100=4
0.1< 0.2|||3325×100=12
0.2< 0.3||||||||||||151525×100=60
0.3< 0.4||||5525×100=20
0.4< 0.5|1125×100=4
Total 25 

Mid point:

The midpoint is the average of the lower limit and upper limit of a particular class. It is also called as class mark.

Midpoint=(Lowerclass limit+Upperclasslimit)2

Thus, the mid points for each class is tabulated below:

Bin limits

Frequency

f

Mid point
LowerUpper
0< 0.110+0.12=0.05
0.1< 0.230.1+0.22=0.15
0.2< 0.3150.2+0.32=0.25
0.3< 0.450.3+0.42=0.35
0.4< 0.510.4+0.52=0.45
Total25 

Cumulative frequency:

Cumulative frequency is the sum of all frequencies up to that class. The last class’s cumulative frequency is equal to the sample size n.

Thus, the cumulative frequency for each calss is tabulated below:

Bin limits

Frequency

f

Cumulative

frequency

LowerUpper
0< 0.111
0.1< 0.231+3=4
0.2< 0.31515+4=19
0.3< 0.4519+5=24
0.4< 0.5124+1=25
Total25 

Cumulative Relative frequency:

Bin limits

Cumulative

frequency

Cumulative

percent

LowerUpper
0< 0.11125×100=4
0.1< 0.24425×100=16
0.2< 0.3191925×100=76
0.3< 0.4242425×100=96
0.4< 0.5252525×100=100
Total  

Software procedure:

Step-by-step software procedure to obtain histogram using MINITAB software is as follows,

  • • Choose Graph > Histogram.
  • • Choose Simple, and then click OK.
  • • In Graph variables, enter the corresponding column of Yankees.
  • • Click Scale>Y-Scale Type> Percent
  • • Click OK.
  • • To modify the interval settings, double click on the horizontal axis of the graph. Then, select Binning>Cutpoint>CutpointPositions,in this box, enter the values for the cut points of the bin intervals (0, 0.1, 0.2, 0.3, 0.4 and 0.5).
  • For 9 bins:

Here the number of bins are considered as 9 with width 0.05.

The frequeny table is as follows,

Bin limitsTally

Frequency

f

Percent
LowerUpper
0< 0.05|1125×100=4
0.05< 0.10 0025×100=0
0.10< 0.15|1125×100=4
0.15< 0.20||2225×100=8
0.20< 0.25|||||6625×100=24
0.25< 0.30||||||||9925×100=36
0.30< 0.35||||5525×100=20
0.35< 0.40 0025×100=0
0.45< 0.45|1125×100=4
Total 25 

Mid point:

The midpoint is the average of the lower limit and upper limit of a particular class. It is also called as class mark.

Midpoint=(Lowerclass limit+Upperclasslimit)2

Thus, the mid points for each class is tabulated below:

Bin limits

Frequency

f

Mid point
LowerUpper
0< 0.0510+0.052=0.025
0.05< 0.1000.05+0.102=0.075
0.10< 0.1510.10+0.152=0.125
0.15< 0.2020.15+0.202=0.175
0.20< 0.2560.20+0.252=0.225
0.25< 0.3090.25+0.302=0.275
0.30< 0.3550.30+0.352=0.325
0.35< 0.4000.35+0.402=0.375
0.40< 0.4510.40+0.452=0.425
Total25 

Cumulative frequency:

Cumulative frequency is the sum of all frequencies up to that class. The last class’s cumulative frequency is equal to the sample size n.

Thus, the cumulative frequency for each calss is tabulated below:

Bin limits

Frequency

f

Cumulative

frequency

LowerUpper
0< 0.0511
0.05< 0.1001+0=1
0.10< 0.1511+1=2
0.15< 0.2022+2=4
0.20< 0.2564+6=10
0.25< 0.30910+9=19
0.30< 0.35519+5=24
0.35< 0.40024+0=24
0.40< 0.45124+1=25
Total25 

Cumulative Relative frequency:

Bin limits

Cumulative

frequency

Cumulative

percent

LowerUpper
0< 0.051125×100=4
0.05< 0.101125×100=4
0.10< 0.152225×100=8
0.15< 0.204425×100=16
0.20< 0.25101025×100=40
0.25< 0.30191925×100=76
0.30< 0.35242425×100=96
0.35< 0.40242425×100=96
0.40< 0.45252525×100=100
Total  

Software procedure:

Step-by-step software procedure to obtain histogram using MINITAB software is as follows,

  • • Choose Graph > Histogram.
  • • Choose Simple, and then click OK.
  • • In Graph variables, enter the corresponding column of Yankees.
  • • Click Scale>Y-Scale Type> Percent
  • • Click OK.
  • • To modify the interval settings, double click on the horizontal axis of the graph. Then, select Binning>Cutpoint>CutpointPositions,in this box, enter the values for the cut points of the bin intervals (0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40 and 0.45).

d.

Expert Solution
Check Mark
To determine

Explain whether the visual impression of the data change when the number of bins increased or not

Answer to Problem 30CE

Yes, the visual impression of the data changes when the number of bins are varied.

Explanation of Solution

From the histograms in part (c), it is observed that when the bins are 5 the histogram appears to be more symmetrical. And when the bins are 9 then, the histogram appears to be skewed left. Hence there is change the appearance of the histogram when the number of bins vary.

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Chapter 3 Solutions

Gen Combo Ll Applied Statistics In Business & Economics; Connect Access Card

Ch. 3.4 - (a) Use Excel to prepare a line chart to display...Ch. 3.4 - Prob. 12SECh. 3.4 - Prob. 13SECh. 3.4 - (a) Use Excel to prepare a line chart to display...Ch. 3.5 - Prob. 15SECh. 3.5 - The table shows the number of TVs sold by a major...Ch. 3.6 - (a) Use Excel to prepare a 2-D pie chart for these...Ch. 3.6 - Prob. 19SECh. 3.6 - Prob. 20SECh. 3.7 - (a) Use Excel to make a scatter plot of the data...Ch. 3.7 - Prob. 22SECh. 3.7 - Prob. 23SECh. 3.7 - (a) Use Excel to make a scatter plot of the data,...Ch. 3 - Name two attractive features and two limitations...Ch. 3 - (a) What is a frequency distribution? (b) What are...Ch. 3 - (a) What is a histogram? (b) What does it show?Ch. 3 - (a) What is a bimodal histogram? (b) Explain the...Ch. 3 - (a) What is a scatter plot? (b) What do scatter...Ch. 3 - For what kind of data would we use a bar chart?...Ch. 3 - For what kind of data would we use a line chart?...Ch. 3 - (a) List the three most common types of charts in...Ch. 3 - (a) For what kind of data would we use a pie...Ch. 3 - Which types of charts can be used for time series...Ch. 3 - (a) When might we need a log scale? (b) What do...Ch. 3 - When might we use a stacked column chart? An area...Ch. 3 - List five deceptive graphical techniques.Ch. 3 - What is a pivot table? Why is it useful?Ch. 3 - The durations (minutes) of 26 electric power...Ch. 3 - The U.S. Postal Service will ship a Priority Mail,...Ch. 3 - A study of 40 U.S. cardiac care centers showed the...Ch. 3 - Prob. 28CECh. 3 - An executives telephone log showed the following...Ch. 3 - Below are batting averages of the New York Yankees...Ch. 3 - Prob. 31CECh. 3 - Prob. 32CECh. 3 - Download the full data set from the McGraw-Hill...Ch. 3 - (a) What kind of display is this? (b) Identify its...Ch. 3 - (a) What kind of display is this? (b) Identify its...Ch. 3 - (a) What kind of display is this? (b) Identify its...Ch. 3 - (a) What kind of display is this? (b) Identify its...Ch. 3 - Prob. 38CECh. 3 - (a) What kind of display is this? (b) Identify its...Ch. 3 - (a) What kind of display is this? (b) Identify its...Ch. 3 - Prob. 41CECh. 3 - Prob. 42CECh. 3 - (a) Use Excel to prepare an appropriate type of...Ch. 3 - Prob. 44CECh. 3 - Prob. 45CECh. 3 - (a) Use Excel to prepare an appropriate type of...Ch. 3 - Prob. 47CECh. 3 - (a) Use Excel to prepare an appropriate type of...
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