The paper "Lessons from Pacemaker Implantations" 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 summarized in the table below. 16 28 31 14 23 33 11 10 22 33 26 16 15 27 29 26 26 7 15 16 5 17 18 6 19 10 6 20 6 18 23 12 24 19 21 24 27 13 23 16 11 34 16 14 18 17 22 25 26 28 26 4 13 28 9 16 29 15 16 7 27 18 21 3 14 22 35 10 14 28 18 25 8 7 5 26 24 25 15 28 9 24 4 10 13 18 19 11 13 (a) Fill in the following table. (Round your answer to four decimal places if needed.) Classes Frequency Relative Frequency Cumulative Relative Frequency 0 - <6 6 - <12 12 - <18 18 - <24 24 - <30 30 - <36 (b) Use the cumulative relative frequencies to give approximate answers to the following questions. (Round your answer to four decimal places if needed.) What proportion of those who participated in the study had pacemakers that did not malfunction within the first year? 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? p = (c) Construct a cumulative relative frequency plot. Prepare an Excel document. The file should be named with a .xls extension. Comment: Use it to answer the following questions. (Round your answers to the nearest whole number if needed.) 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?
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
The paper "Lessons from Pacemaker Implantations" 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 summarized in the table below.
|
16 |
28 |
31 |
14 |
23 |
33 |
11 |
10 |
22 |
33 |
26 |
16 |
|
15 |
27 |
29 |
26 |
26 |
7 |
15 |
16 |
5 |
17 |
18 |
6 |
|
19 |
10 |
6 |
20 |
6 |
18 |
23 |
12 |
24 |
19 |
21 |
24 |
|
27 |
13 |
23 |
16 |
11 |
34 |
16 |
14 |
18 |
17 |
22 |
25 |
|
26 |
28 |
26 |
4 |
13 |
28 |
9 |
16 |
29 |
15 |
16 |
7 |
|
27 |
18 |
21 |
3 |
14 |
22 |
35 |
10 |
14 |
28 |
18 |
25 |
|
8 |
7 |
5 |
26 |
24 |
25 |
15 |
28 |
9 |
24 |
4 |
10 |
|
13 |
18 |
19 |
11 |
13 |
(a) Fill in the following table. (Round your answer to four decimal places if needed.)
|
Classes |
Frequency |
Relative Frequency |
Cumulative Relative Frequency |
|
0 - <6 |
|||
|
6 - <12 |
|||
|
12 - <18 |
|||
|
18 - <24 |
|||
|
24 - <30 |
|||
|
30 - <36 |
(b) Use the cumulative relative frequencies to give approximate answers to the following questions. (Round your answer to four decimal places if needed.)
- What proportion of those who participated in the study had pacemakers that did not malfunction within the first year?
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?
p =
(c) Construct a cumulative relative frequency plot. Prepare an Excel document. The file should be named with a .xls extension.
Comment:
Use it to answer the following questions. (Round your answers to the nearest whole number if needed.)
- 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 stillfunctioning ?
Part a:
From the provided data we the table as below:
|
Classes |
Frequency |
Relative frequency |
Cumulative relative frequency |
|
0 - < 6 |
5 |
5/89 = 0.0562 |
0.0562 |
|
6 - < 12 |
16 |
16/89 = 0.1798 |
0.2360 |
|
12 - < 18 |
22 |
22/89 = 0.2472 |
0.4831 |
|
18 - < 24 |
18 |
18/89 = 0.2022 |
0.6854 |
|
24 - < 30 |
23 |
23/89 = 0.2584 |
0.9438 |
|
30 - < 36 |
5 |
5/89 = 0.0562 |
1.0000 |
|
Sum |
89 |
1 |
|
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