Chapter 3, Problem 12CRE

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:

1. a. Summarize these data in the form of a frequency distribution, using class intervals of 0 to <6, 6 to <12, and so on.
2. b. Calculate the relative frequencies and cumulative relative frequencies for each class interval of the frequency distribution of Part (a).
3. c. Show how the relative frequency for the class interval 12 to <18 could be obtained from the cumulative relative frequencies.
4. d. Use the cumulative relative frequencies to give approximate answers to the following:
   1. i. What proportion of those who participated in the study had pacemakers that did not malfunction within the first year?
   2. ii. If the pacemaker must be replaced as soon as the first electrical malfunction occurs, approximately what proportion required replacement between 1 and 2 years after implantation?
5. e. Construct a cumulative relative frequency plot, and use it to answer the following questions.
   1. i. What is the approximate time at which 50% of the pacemakers had failed?
6. ii. What is the approximate time at which only 10% of the pacemakers initially implanted were still functioning?

To determine

Construct the frequency distribution for the given data.

Answer to Problem 12CRE

The frequency distribution is given below.

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

Explanation of Solution

Calculation:

The data represents the time to the first electrical malfunction of the pacemaker for 89 heart patients.

Software procedure:

Frequency distribution:

The number of values lying in the particular interval or the number of times each value repeats is the frequency of that particular class interval or event.

The frequencies are calculated by using the tally mark. Here, the number of times each activity repeats is the frequency of that particular physical activity.

Here, the number of values time under the specified interval is the frequency of that particular class interval of time.

Here, the number of values in between the class interval 0-<6 is 2.

Therefore, the frequency of the class interval 0-<6 is 2.

Similarly, the frequencies of all the remaining class intervals are as follows:

Class interval	Tally	Frequency
0-<6	$\left|\right|$	2
6-<12	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}$	10
12-<18	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}|$	21
18-<24	$\bcancel{\left|\right|\left|\right|}\left|\right||$	28
24-<30	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\left|\right|$	22
30-<36	$\bcancel{\left|\right|\left|\right|}|$	6
Total		89

To determine

Construct the relative frequency distribution for the given data.

Construct the cumulative relative frequency distribution for the given data.

Answer to Problem 12CRE

The relative frequency distribution and cumulative relative frequency distribution are given below.

Class interval	Relative frequency	Cumulative relative frequency
0-<6	0.02247	0.02247
6-<12	0.11236	0.13483
12-<18	0.23596	0.37079
18-<24	0.31461	0.6854
24-<30	0.24719	0.39259
30-<36	0.06742	1

Explanation of Solution

Calculation:

The general formula to obtain the relative frequency is given below:

$\text{Relative}\text{Frequency = }\frac{\text{Frequency}}{\text{Total frequency}}$

Substitute the frequency of the class interval 0-<6 as “2” and the total frequency as “89” in relative frequency.

$\begin{array}{c}\text{Relative}\text{Frequency}\text{of the class interval 0-<6}=\frac{2}{89}\\ =0.02247\end{array}$

Similarly, relative frequencies for the remaining class intervals are obtained below:

Class interval	Frequency	Relative frequency
0-<6	2	$\frac{2}{89}=0.02247$
6-<12	10	$\frac{10}{89}=0.11236$
12-<18	21	$\frac{21}{89}=0.23596$
18-<24	28	$\frac{28}{89}=0.31461$
24-<30	22	$\frac{22}{89}=0.24719$
30-<36	6	$\frac{6}{89}=0.06742$

Cumulative relative frequency:

Cumulative relative frequency is the sum of relative frequencies of all the previous events which are arranged in an order from smallest to largest value.

The general formula to obtain cumulative frequency using frequency distribution is,

$\left(\begin{array}{l}\text{Cumulative relative frequency}\\ \text{of the present event }\end{array}\right)\text{= }\left(\begin{array}{l}\text{Relative frequency }\\ \text{of present event}\end{array}\right)+\left(\begin{array}{l}\text{Cumulative relative frequency}\\ \text{of immediate}\text{preceding event}\end{array}\right)$

From the relative frequencies, the cumulative relative frequencies are obtained as follows:

Class interval	Relative frequency	Cumulative relative frequency
0-<6	0.02247	$0.02247+0=0.02247$
6-<12	0.11236	$0.02247+0.11236=0.13483$
12-<18	0.23596	$0.13483+0.23596=0.37079$
18-<24	0.31461	$0.37079+0.31461=0.6854$
24-<30	0.24719	$0.6854+0.24719=0.93259$
30-<36	0.06742	$0.93259+0.06742=1$

To determine

Obtain the relative frequency for the class interval 12-18 using the cumulative frequency distribution.

Answer to Problem 12CRE

The relative frequency for the class interval 12-18 using the cumulative frequency distribution is 0.23596.

Explanation of Solution

Calculation:

The general formula to obtain cumulative frequency using frequency distribution is given below:

$\left(\begin{array}{l}\text{Cumulative relative frequency}\\ \text{of the present event }\end{array}\right)\text{= }\left(\begin{array}{l}\text{Relative frequency }\\ \text{of present event}\end{array}\right)+\left(\begin{array}{l}\text{Cumulative relative frequency}\\ \text{of immediate}\text{preceding event}\end{array}\right)$

Relative frequency of a present event is obtained using the formula given below:

$\left(\begin{array}{l}\text{Relative frequency }\\ \text{of present event}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Cumulative relative frequency}\\ \text{of the present event }\end{array}\right)-\left(\begin{array}{l}\text{Cumulative relative frequency}\\ \text{of immediate}\text{preceding event}\end{array}\right)$

From the cumulative relative frequency distribution, relative frequency distribution is obtained as given below:

$\begin{array}{c}\left(\begin{array}{l}\text{Relative frequency }\\ \text{of the class interval}\\ \text{12-<18}\end{array}\right)\text{= }\left(\begin{array}{l}\text{Cumulative relative frequency}\\ \text{of the class interval 12-<18 }\end{array}\right)-\left(\begin{array}{l}\text{Cumulative relative frequency}\\ \text{of the class interval 6-<12}\end{array}\right)\\ =0.37079-0.13483\\ =0.23596\end{array}$

Thus, the relative frequency for the class interval 12-18 using the cumulative frequency distribution is 0.23596.

To determine

(i). Find the approximate proportion of heart patients who had pacemakers that did not malfunction within the first year.

(ii) Find the approximate proportion of heart patients who required replacement between 1 and 2 years after implantation.

Answer to Problem 12CRE

(i) The approximate proportion of heart patients who had pacemakers that did not malfunction within the first year is 0.86517.

(ii) The approximate proportion of heart patients who required replacement between 1 and 2 years after implantation is 0.55057.

Explanation of Solution

Calculation:

The general formula for the relative frequency or proportion is,

$\text{Relative}\text{Frequency or proportion = }\frac{\text{Frequency}}{\text{Total frequency}}$

(i). Approximate proportion of heart patients who had pacemakers that did not malfunction within the first year:

The Objective is to find the cumulative relative frequency of heart patients who had pacemakers that did not malfunction within the first year.

The cumulative relative frequency of heart patients who had pacemakers that malfunction within the first year is 0.13483.

That is, the proportion of heart patients who had pacemakers that malfunction within the first year is 0.13483.

Hence, the cumulative relative frequency of heart patients who had pacemakers that did not malfunction within the first year is obtained as given below:

$1-0.13483=0.86517$

Thus, the approximate proportion of heart patients who had pacemakers that did not malfunction within the first year is 0.86517.

(ii). Approximate proportion of heart patients who required replacement between 1 and 2 years after implantation:

The Objective is to find the relative frequency of heart patients required replacement between 1 and 2 years after implantation.

The relative frequency of heart patients required replacement between 1 and 2 years after implantation is obtained as given below:

$0.23596+0.31461=0.55057$

Thus, the approximate proportion of heart patients who required replacement between 1 and 2 years after implantation is 0.55057.

To determine

Plot the cumulative frequency distribution for the given data.

(i) Find the approximate time at which 50% of the pacemakers had failed.

(ii) Find the approximate time at which only 10% of the initially implanted pacemakers are functioning.

Answer to Problem 12CRE

Cumulative distribution plot is given below:

(i) The time at which 50% of the pacemakers had failed will be in between 18-<24 months.

(ii) The approximate time at which only 10% of the initially implanted pacemakers are functioning will be in between 24-<30 months.

Explanation of Solution

Calculation:

The cumulative relative frequency histogram is plotted for the given data.

Procedure to plot cumulative distribution plot:

Step by step procedure to draw the cumulative distribution plot is given below.

• Draw a horizontal axis and a vertical axis.
• The horizontal axis represents the cumulative frequencies.
• The vertical axis represents the “Time to first malfunction in months”.
• Plot each of the 6 cumulative frequencies corresponding to the time to first malfunction in months.
• Connect all the 6 plotted points of cumulative frequencies with a line.

The general formula for the relative frequency or proportion is,

$\text{Relative}\text{Frequency or proportion = }\frac{\text{Frequency}}{\text{Total frequency}}$

(i). Approximate time at which 50% of the pacemakers had failed:

The Objective is to find the time at which 50% of the pacemakers had failed.

From the cumulative relative frequency distribution, 0.6854 corresponds to the interval 18-<24 months.

The cumulative relative frequency of 0.5 is less than the cumulative relative frequency of 0.6854.

Thus, the time at which 50% of the pacemakers had failed will be in between 18-<24 months.

(ii). Approximate time at which only 10% of the initially implanted pacemakers are functioning:

The Objective is to find the time only 10% of the initially implanted pacemakers are functioning.

In other words it can be said that, the time at which 90% of the pacemakers had failed.

From the cumulative relative frequency distribution, 0.93259 corresponds to the interval 24-<30 months.

The cumulative relative frequency of 0.9 is less than the cumulative relative frequency of 0.93259.

Thus, the time at which only 10% of the initially implanted pacemakers are functioning will be in between 24-<30 months.