Chapter 2, Problem 52SE

A survey of commercial buildings served by the Cincinnati Gas & Electric Company asked what main heating fuel was used and what year the building was constructed. A partial crosstabulation of the findings follows.

	Fuel Type
Year Constructed	Electricity	Natural Gas	Oil	Propane	Other
1973 or before	40	183	12	5	7
1974–1979	24	26	2	2	0
1980–1986	37	38	1	0	6
1987–1991	48	70	2	0	1

1. a. Complete the crosstabulation by showing the row totals and column totals.
2. b. Show the frequency distributions for year constructed and for fuel type.
3. c. Prepare a crosstabulation showing column percentages.
4. d. Prepare a crosstabulation showing row percentages.
5. e. Comment on the relationship between year constructed and fuel type.

To determine

Complete the cross tabulation by showing the row totals and column totals.

Answer to Problem 52SE

The completed cross tabulation is,

	Fuel Type					Total
Year Constructed	Electricity	Natural Gas	Oil	Propane	Other
1973 or more	40	183	12	5	7	247
1974-1979	24	26	2	2	0	54
1980-1986	37	38	1	0	6	82
1987-1991	48	70	2	0	1	121
Total	149	317	17	7	14	504

Explanation of Solution

Calculation:

The given information is that a partial cross tabulation of fuel type and year constructed.

The row total and column total can be obtained as follows:

$\begin{array}{c}\text{Row total for the year}\left(\text{1973 or more}\right)\text{ = 40+183+12+5+7}\\ \text{= 247}\end{array}$

$\begin{array}{c}\text{Column total of fuel type (Electricity)}=40+24+37+48\\ =149\end{array}$

The remaining row and column total can be obtained as follows:

	Fuel Type					Total
Year Constructed	Electricity	Natural Gas	Oil	Propane	Other
1973 or more	40	183	12	5	7	247
1974-1979	24	26	2	2	0	54
1980-1986	37	38	1	0	6	82
1987-1991	48	70	2	0	1	121
Total	149	317	17	7	14	504

To determine

Construct a frequency distribution for year constructed and for fuel type.

Answer to Problem 52SE

The frequency distribution for year constructed is,

Year Constructed	Frequency
1973 or more	247
1974–1979	54
1980–1986	82
1987–1991	121
Total	504

The frequency distribution for fuel type is,

Fuel Type	Frequency
Electricity	149
Natural Gas	317
Oil	17
Propane	7
Other	14
Total	504

Explanation of Solution

Calculation:

For year constructed:

The frequencies are calculated by using the tally mark for each class interval.

• The class intervals are “1973 or more”, “1974–1979”, “1980–1986” and “1987–1991”.
• Make a tally mark for each value of year constructed in the corresponding class interval and continue for all values in the data.
• The number of tally marks in each class represents the frequency, f of that class.

The frequency distribution for year constructed is obtained as given below:

Year Constructed	Tally marks	Frequency
1973 or more	$\begin{array}{l}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\\ \bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\left|\right|\end{array}$	247
1974–1979	$\begin{array}{l}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\\ \bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\left|\right|\left|\right|\end{array}$	54
1980–1986	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\left|\right|$	82
1987–1991	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}|$	121
Total		504

For fuel type:

Here, “electricity” is the type of fuel which is repeated for 149 times in the data set and thus 149 is the frequency for the fuel type electricity.

Similarly, the frequency of remaining company size is given below:

Fuel Type	Frequency
Electricity	149
Natural Gas	317
Oil	17
Propane	7
Other	14
Total	504

To determine

Compute a cross tabulation showing column percentages for the data.

Answer to Problem 52SE

Cross tabulation of column percentages is:

	Fuel Type
Year Constructed	Electricity	Natural Gas	Oil	Propane	Other
1973 or more	26.85	57.73	70.59	71.43	50
1974–1979	16.11	8.2	11.76	28.57	0
1980–1979	24.83	11.99	5.88	0	42.86
1987–1991	32.21	22.08	11.76	0	7.14
Total	100	100	100	100	100

Explanation of Solution

Calculation:

Software procedure:

Step-by-step procedure to construct the column percentage using MINITAB software is given below:

• Choose stat>tables>cross tabulation and chi square.
• Choose raw data (summarized).
• In rows, enter the column of year constructed.
• In columns, enter the column of fuel type.
• Under display, select column percent.
• Click OK

Output using the MINITAB software is given below:

To determine

Compute the cross tabulation showing row percentages for the data.

Answer to Problem 52SE

	Fuel Type					Total
Year Constructed	Electricity	Natural Gas	Oil	Propane	Other
1973 or more	16.19	74.09	4.86	2.02	2.83	100
1974–1979	44.44	48.15	3.7	3.7	0	100
1980–1979	45.12	46.34	1.22	0	7.32	100
1987–1991	39.67	57.85	1.65	0	0.83	100

Explanation of Solution

Calculation:

Software procedure:

Step-by-step procedure to construct the row percentage using MINITAB software is given below:

• Choose stat>tables>cross tabulation and chi square.
• Choose raw data (summarized).
• In rows, enter the column of year constructed.
• In columns, enter the column of fuel type.
• Under display, select row percent.
• Click OK

Output using the MINITAB software is given below:

To determine

Comment on the relationship between year constructed and fuel type.

Explanation of Solution

Observations from the column percentages cross tabulation:

The percentage of buildings using electricity has not changed greatly over the years. Most of the buildings using natural gas were constructed in 1973 or before. Also for the buildings using oil most of them were constructed in 1973 or before. The percentage of buildings using propane was constructed in 1973 or before.

Observations from the row percentages cross tabulation:

Most of the building was constructed using electricity or natural gas. In 1973 or before most of the buildings used natural gas. In the period 1974–1979, most of the building was constructed using electricity and natural gas. From 1987 onwards most of the building was constructed using natural gas and second largest percentage was constructed using electricity.