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: \(\begin{array}{rrrrrrrrrrrr}24 & 20 & 16 & 32 & 14 & 22 & 2 & 12 & 24 & 6 & 10 & 20 \\ 8 & 16 & 12 & 24 & 14 & 20 & 18 & 14 & 16 & 18 & 20 & 22 \\ 24 & 26 & 28 & 18 & 14 & 10 & 12 & 24 & 6 & 12 & 18 & 16 \\ 34 & 18 & 20 & 22 & 24 & 26 & 18 & 2 & 18 & 12 & 12 & 8 \\ 24 & 10 & 14 & 16 & 22 & 24 & 22 & 20 & 24 & 28 & 20 & 22 \\ 26 & 20 & 6 & 14 & 16 & 18 & 24 & 18 & 16 & 6 & 16 & 10 \\\ 14 & 18 & 24 & 22 & 28 & 24 & 30 & 34 & 26 & 24 & 22 & 28 \\ 30 & 22 & 24 & 22 & 32 & & & & & & & \end{array}\) 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 frequency distribution of Part (a). c. Show how the relative frequency for the class interval 12 to \(<18\) could be obtained from the cumulative relative frequencies. d. Use the cumulative relative frequencies to give approximate answers to the following: i. What proportion of those who participated in the study had pacemakers that did not malfunction within the first year? 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? e. Construct a cumulative relative frequency plot, and use it to answer the following questions. i. What is the approximate time at which about \(50 \%\) of the pacemakers had failed? ii. What is the approximate time at which only about \(10 \%\) of the pacemakers initially implanted were still functioning?

Short answer

The detailed solutions for frequency distributions, relative frequencies and cumulative relative frequencies will vary based on calculations. Use the cumulative relative frequencies to infer proportions of pacemakers and interpret their functionality as described in step 4 and 5.

Step-by-step solution

A: Construct Frequency Distribution

Firstly, the data given needs to be formatted into a frequency distribution using class intervals of 0 to < 6,6 to < 12 and so on.

B: Compute Relative Frequencies and Cumulative Relative Frequencies

Next, using the frequency distribution from step 1, for each class interval, compute the relative frequencies (frequency in each interval / total number of data points) and cumulative relative frequencies, by accumulating the relative frequency of each class interval.

C: Calculate Relative Frequency using Cumulative Relative Frequencies

For the class interval 12 to <18, the relative frequency can be obtained by subtracting the cumulative relative frequency of the previous class interval from the cumulative relative frequency of this interval.

D: Answer Questions based on Cumulative Relative Frequencies

To determine the proportion of individuals within the first year without a malfunction, refer to the cumulative relative frequency corresponding to 12. For the pacemakers needing replacement in 1-2 years, subtract the cumulative relative frequency at 12 from that at 24.

E: Construct Cumulative Relative Frequency Plot and Interpret

Create a plot of the cumulative relative frequencies against the class intervals. Use this to determine and interpret the time at which ~50% of the pacemakers failed and at which ~10% of the pacemakers are still functioning.