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Exercise 1

Weights (g) In column A are listed the weights (in grams) of 40 silver quarters minted before 1964.
6.2771
6.2371
6.1501 1 Sort the data in ascending order.
6.0002 2 Compute the range. range = Max - Min
6.1275 3 Estimate the standard deviation using the range rule of thumb.
6.2866 4 Range rule of thumb: SD is approximate range/4
6.076 5 Use Sturgis's formula to find the ideal number of classes.
6.1426 6 Find an appropriate class width.
6.3415 7 Choose a starting value and create all lower limits.
6.1309 8 Now fill in the upper limits.
6.2151 9 Use EXCEL to find the frequencies for each class.
6.2412 10 Compute the relative frequencies.
6.1442 11 Compute the percent frequencies.
6.1073 12 Construct a histogram.
6.1181 13 State whether the distribution is roughly bell-shaped (overall low-high-low pattern), roughly uniform, left skewed, or right skewed.
6.1352 14 Use EXCEL to find the summary statistics.
6.2821 15 Find the coefficient of variation. CV = SD/mean
6.2647
6.2908
6.1661
6.1947
6.194
6.0257
6.1719
6.3278
6.2674
6.2718
6.1949
6.2465
6.3172
6.1487
6.0829
6.1423
6.197
6.2441
6.3669
6.0775
6.1095
6.1787
6.213

Solution 1

Weights (g)
6.0002 range 0.3667
6.0257 approx SD 0.09168
6.076
6.0775 To construct a frequency distribution:
6.0829 Find the optimal # of classes = 1+3.3*log(n) 6
6.1073 Find the class width =range/# of classes 0.0611166667 round up 0.07
6.1095
6.1181 Lower limits Upper limits Midpoints FRQ REL FRQ % Upper limits Frequency
6.1275 6.0000 6.0699 6.03495 2 0.05 5 6.0699 2
6.1309 6.0700 6.1399 6.10495 9 0.225 22.5 6.1399 9
6.1352 6.1400 6.2099 6.17495 12 0.3 30 6.2099 12
6.1423 6.2100 6.2799 6.24495 10 0.25 25 6.2799 10
6.1426 6.2800 6.3499 6.31495 6 0.15 15 6.3499 6
6.1442 6.3500 6.4199 6.38495 1 0.025 2.5 6.4199 1
6.1487 40 1 100
6.1501
6.1661
6.1719
6.1787
6.194
6.1947
6.1949
6.197
6.213
6.2151
6.2371
6.2412
6.2441
6.2465
6.2647
6.2674
6.2718
6.2771 This histogram is roughly bell-shaped with the low-high-low pattern.
6.2821
6.2866
6.2908
6.3172
6.3278
6.3415
6.3669

WEIGHTS OF PRE-1964 QUARTERS

RELATIVE FREQUENCY 6.0349500000000003 6.1049500000000005 6.1749500000000008 6.2449500000000011 6.3149500000000014 6.3849500000000017 0.05 0.22500000000000001 0.3 0.25 0.15 2.5000000000000001E-2

CLASS MIDPOINTS

RELATIVE FREQUENCIES

EX1 SUMMARY STATS

Weights (g)
Mean 6.19267
Standard Error 0.013755123
Median 6.19435
Mode ERROR:#N/A There were no repeated values.
Standard Deviation 0.08700
Sample Variance 0.00757
Kurtosis -0.5032693808
Skewness -0.0501955057
Range 0.3667 CV 0.01405
Minimum 6.0002
Maximum 6.3669
Sum 247.7069
Count 40

Exercise 2

Weights (g) In column A are listed the weights (in grams) of 40 silver quarters minted after 1964.
5.7027
5.7495
5.705 1 Sort the data in ascending order.
5.5941 2 Compute the range. range = Max - Min
5.7247 3 Estimate the standard deviation using the range rule of thumb.
5.6114 4 Range rule of thumb: SD is approximate range/4
5.616 5 Use Sturgis's formula to find the ideal number of classes.
5.5999 6 Find an appropriate class width.
5.779 7 Choose a starting value and create all lower limits.
5.6841 8 Now fill in the upper limits.
5.6234 9 Use EXCEL to find the frequencies for each class.
5.5928 10 Compute the relative frequencies.
5.6486 11 Compute the percent frequencies.
5.6661 12 Construct a histogram.
5.5361 13 State whether the distribution is roughly bell-shaped (overall low-high-low pattern), roughly uniform, left skewed, or right skewed.
5.5491 14 Use EXCEL to find the summary statistics.
5.7239 15 Find the coefficient of variation. CV = SD/mean
5.6555
5.6063
5.5709
5.5591
5.5864
5.6694
5.5454
5.6646
5.6872
5.6274
5.6157
5.6668
5.7198
5.5636
5.6485
5.6703
5.6848
5.5609
5.7344
5.6449
5.5804
5.601
5.6022

Solution 2

Weights (g)
5.5361 RANGE 0.2429
5.5454 approx SD 0.060725
5.5491
5.5591 To construct a frequency distribution:
5.5609 Find the optimal # of classes = 1+3.3*log(n) 6
5.5636 Find the class width =range/# of classes 0.0404833333 round up 0.05
5.5709
5.5804 Lower limits Upper limits Midpoints FRQ REL FRQ % Upper limits Frequency
5.5864 5.5000 5.5499 5.52495 3 0.075 7.5 5.5499 3
5.5928 5.5500 5.5999 5.57495 9 0.225 22.5 5.5999 9
5.5941 5.6000 5.6499 5.62495 11 0.275 27.5 5.6499 11
5.5999 5.6500 5.6999 5.67495 9 0.225 22.5 5.6999 9
5.601 5.7000 5.7499 5.72495 7 0.175 17.5 5.7499 7
5.6022 5.7500 5.7999 5.77495 1 0.025 2.5 5.7999 1
5.6063 40 1 100
5.6114
5.6157
5.616
5.6234
5.6274
5.6449
5.6485
5.6486
5.6555
5.6646
5.6661
5.6668
5.6694
5.6703
5.6841
5.6848
5.6872
5.7027
5.705 This distribution is roughly bell-shaped with the low-high-low pattern.
5.7198
5.7239
5.7247
5.7344
5.7495
5.779

WEIGHTS OF POST-1964 QUARTERS

RELATIVE FREQUENCY 5.5249500000000005 5.5749499999999994 5.6249500000000001 5.6749499999999991 5.7249499999999998 5.7749499999999987 7.4999999999999997E-2 0.22500000000000001 0.27500000000000002 0.22500000000000001 0.17499999999999999 2.5000000000000001E-2

MIDPOINTS

RELATIVE FREQUENCY

EX2 SUMMARY STATS

Weights (g)
Mean 5.63930
Standard Error 0.009793063
Median 5.63615
Mode ERROR:#N/A There are no repeated values.
Standard Deviation 0.06194
Sample Variance 0.00384
Kurtosis -0.7309998522
Skewness 0.2740343064
Range 0.24290 CV 0.01098
Minimum 5.5361
Maximum 5.779
Sum 225.5719
Count 40

Exercise 3

Which of the data sets, the pre-1964 quarters or post-1964 quarters, has the most internal variation?
Use the coefficient of variation. The greater the coefficient of variation, the greater the internal variation.

Solution 3

Which of the data sets, the pre-1964 quarters or post-1964 quarters, has the most internal variation?
Use the coefficient of variation. The greater the coefficient of variation, the greater the internal variation.
The CV of the pre-1964 quarters is 0.01405.
The CV of the post-1964 quarters is 0.01098.
The pre-1964 quarters have more internal variation since they have a greater CV.