Question

The paper “Lessons from Pacemaker Implantations” (Journal of the American Medical Association [1965]:231–232) gave the results of a study that followed 89 heart patients who had received electronic pacemakers.The time (in months) to the first electrical malfunction of the pacemaker was recorded: 24, 20, 16, 32, 14, 22,2, 12, 24, 6, 10, 20, 8, 16, 12, 24, 14, 20, 18, 14, 16, 18, 20, 22, 24, 26, 28, 18, 14, 10, 12, 24, 6, 12, 18, 16, 34,18, 20, 22, 24, 26, 18, 2, 18, 12, 12, 8, 24, 10, 14, 16, 22, 24, 22, 20, 24, 28, 20, 22, 26, 20, 6, 14, 16, 18, 24, 18,16, 6, 16, 10, 14, 18, 24, 22, 28, 24, 30, 34, 26, 24, 22, 28, 30, 22, 24, 22, 32.
a) Summarize these data in the form of a frequency distribution, using class intervals of 0 to <6, 6 to <12,and so on.
b) Compute the relative frequencies and cumulative relative frequencies for each class interval of the frequencydistribution of Part (a).
c) Show how the relative frequency for the class interval 12 to <18 could be obtained from the cumulativerelative frequencies.
d) Use the cumulative relative frequencies to give approximate answers to the following:
• What proportion of those who participated in the study had pacemakers that did not malfunctionwithin the first year?
• If the pacemaker must be replaced as soon as the first electrical malfunction occurs, approximatelywhat proportion required replacement between 1 and 2 years after implantation?
e) Construct a cumulative relative frequency plot, and use it to answer the following questions.• What is the approximate time at which about 50% of the pacemakers had failed?• What is the approximate time at which only about 10% of the pacemakers initially implanted were still
functioning?

Answer 1

a)

The frequency within each class is obtained in excel using the function =COUNTIF(). The screenshot is shown below,

Class	frequency
0-<6	2
6-<12	10
12-<18	21
18-<24	28
24-<30	22
30-<36	6

b)

The relative frequency is obtained by dividing each class frequency by total frequency

and the cumulative frequency is obtained by adding the current and previous class frequencies.

Class	frequency	Relative frequency	Cumulative frequency
0-<6	2	0.0225	2
6-<12	10	0.1124	12
12-<18	21	0.2360	33
18-<24	28	0.3146	61
24-<30	22	0.2472	83
30-<36	6	0.0674	89
Total	89

c)

d)

e)

The cumulative relative frequency is obtained by dividing the cumulative frequency by the total frequency.

Class	Cumulative frequency	Cumulative relative frequency	Cumulative relative frequency (%)
0-<6	2	0.0225	2.25
6-<12	12	0.1348	13.48
12-<18	33	0.3708	37.08
18-<24	61	0.6854	68.54
24-<30	83	0.9326	93.26
30-<36	89	1	100

The plot between class and cumulative relative frequency is obtained in excel. The screenshot is shown below,

About 50% of the pacemakers had failed between 18 to 24 months (approximately 20th month).

At the time around 30 months, about 10% of the pacemakers initially implanted were still
functioning