/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 The following data are derived f... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 51

The following data are derived from the Monthly Vital Statistics Report (October 1999 ) issued by the National Center for Health Statistics [10]. These data are pertinent to livebirths only.Suppose that infants are classified as low birthweight if they have a birthweight \(<2500 \mathrm{g}\) and as normal birthweight if they have a birthweight \(\geq 2500\) g. Suppose that infants are also classified by length of gestation in the following five categories: <28 weeks, 28-31 weeks, 32-35 weeks, 36 weeks, and \(\geq 37\) weeks. Assume the probabilities of the different periods of gestation are as given in Table \(3.8 .\)Also assume that the probability of low birthweight is .949 given a gestation of \(<28\) weeks, .702 given a gestation of \(28-31\) weeks, .434 given a gestation of \(32-35\) weeks, .201 given a gestation of 36 weeks, and .029 given a gestation of \(\geq 37\) weeks.3.51 Show that the events \\{length of gestation \(\leq 31\) weeks \(\mathrm{~ a n d ~ \\{ l o w ~ b i r t h w e i g h t \\} ~ a r e ~ n o t ~ i n d e p e n d e n t . ~}\).$$\begin{array}{lc} \hline \text { Length of gestation } & \text { Probability } \\ \hline<28 \text { weeks } & .007 \\\28-31 \text { weeks } & .012 \\\32-35 \text { weeks } & .050 \\\36 \text { weeks } & .037 \\\\\geq 37 \text { weeks } & .893\\\\\hline\end{array}$$

Short Answer

Expert verified
The events are not independent because the conditional probability differs from the overall probability.

Step by step solution

01

Identify the Probability of Gestation 鈮� 31 Weeks

The probability of an infant having a length of gestation of less than or equal to 31 weeks is the sum of the probabilities of gestation periods classified as "<28 weeks" and "28-31 weeks." From the table, these probabilities are 0.007 and 0.012, respectively. Thus, the combined probability is: \[ P(\text{Gestation } \leq 31 \text{ weeks}) = P(<28 \text{ weeks}) + P(28-31 \text{ weeks}) = 0.007 + 0.012 = 0.019 \]
02

Calculate the Probability of Low Birthweight Given Gestation 鈮� 31 Weeks

We need the conditional probability of having a low birthweight given that the gestation period is less than or equal to 31 weeks. This is obtained by using probabilities specific to each gestation category: \[ P(\text{Low Birthweight } | \text{Gestation } \leq 31 \text{ weeks}) = \frac{0.007 \times 0.949 + 0.012 \times 0.702}{0.019} \] Calculate this value step-by-step:- Calculate the weighted probabilities: - For "<28 weeks": \(0.007 \times 0.949 = 0.006643\) - For "28-31 weeks": \(0.012 \times 0.702 = 0.008424\)- Sum them up: \(0.006643 + 0.008424 = 0.015067\)- Find the conditional probability: \[ P(\text{Low Birthweight } | \text{Gestation } \leq 31 \text{ weeks}) = \frac{0.015067}{0.019} \approx 0.792 \]
03

Calculate the Probability of Low Birthweight Across All Gestation

We need the total probability for any infant to be low birthweight, across all gestation periods. Calculate this as follows:\[ P(\text{Low Birthweight}) = 0.007 \times 0.949 + 0.012 \times 0.702 + 0.050 \times 0.434 + 0.037 \times 0.201 + 0.893 \times 0.029 \]Breakdown of each term:- "<28 weeks": \(0.007 \times 0.949 = 0.006643\)- "28-31 weeks": \(0.012 \times 0.702 = 0.008424\)- "32-35 weeks": \(0.050 \times 0.434 = 0.0217\)- "36 weeks": \(0.037 \times 0.201 = 0.007437\)- "\geq 37 weeks": \(0.893 \times 0.029 = 0.025897\)Sum all terms:\[ P(\text{Low Birthweight}) = 0.070101 \]
04

Determine Independence by Comparing Conditional and Total Probabilities

The events \{length of gestation \(\leq 31\) weeks\} and \{low birthweight\} are independent if \[ P(\text{Low Birthweight } | \text{Gestation } \leq 31 \text{ weeks}) = P(\text{Low Birthweight}) \]. From Step 2 and Step 3, we have:\[ P(\text{Low Birthweight } | \text{Gestation } \leq 31 \text{ weeks}) \approx 0.792 \] and \[ P(\text{Low Birthweight}) = 0.070101 \]. Since these values are not equal, these events are not independent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory is the branch of mathematics that deals with the likelihood of different outcomes. It helps us understand how likely an event is to occur. In this exercise, we're dealing with the probability of certain health statistics, like birth weight and gestation periods.

The basic idea is to figure out the chance of a specific event happening from all possible events. For example, we calculated the probability that a baby will have a gestation period of 31 weeks or less by adding probabilities from two categories: less than 28 weeks and 28-31 weeks. By using the formula:
  • \[ P( ext{Gestation } \leq 31\text{ weeks}) = P(<28\text{ weeks}) + P(28-31\text{ weeks}) = 0.007 + 0.012 = 0.019 \]
This is a pretty straightforward application of probability theory, showing how probabilities of distinct events add up to create a total probability.
Conditional Probability
Conditional probability is about calculating the probability of an event, given that we know another event has happened. This is a crucial tool in biostatistics, as seen in our problem where we want the probability of an infant being low birthweight given a certain gestation period.

The formula for conditional probability is:
  • \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
In our scenario, we used it to determine the probability that a baby is low birthweight, given they were born at or before 31 weeks of gestation:
  • \[ P(\text{Low Birthweight} \mid \text{Gestation } \leq 31\text{ weeks}) = \frac{0.006643 + 0.008424}{0.019} \approx 0.792 \]
This conditional perspective gives us much more detailed information, especially when specific conditions like premature birth can heavily influence outcomes.
Gestation Periods
Gestation periods refer to the time period from conception to birth. It informs various aspects of health at birth, like birth weight. In this exercise, gestation periods are divided into categories: less than 28 weeks, 28-31 weeks, 32-35 weeks, 36 weeks, and 37 weeks or more. Each category has a probability associated with it based on health statistics data.

Understanding gestation periods helps in assessing the health risks associated with premature births. For instance, babies born before 28 weeks are much more likely to be of low birthweight compared to those born after 37 weeks, as reflected in our conditional probabilities:
  • Low birthweight probability for
    • \( <28 \text{ weeks} \): 0.949
    • \( 28-31 \text{ weeks} \): 0.702
    • \( \geq 37 \text{ weeks} \): 0.029
    This illustrates how crucial the understanding of gestation periods is to medical professionals and researchers.

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Most popular questions from this chapter

The familial aggregation of respiratory disease is a wellestablished clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is \(.15 ;(2)\) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is \(.13 ;(3)\) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is \(.05 ;\) and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04.Suppose the smoking habits of the parents are independent and the probability that the mother is a current smoker is \(.4,\) whereas the probability that the father is a current smoker is .5. What is the probability that both the father and mother are current smokers?

Consider a family with a mother, father, and two children. Let \(A_{1}=\left\\{\text { mother has influenzal, } A_{2}=\\{ \text { father has influenzal, }\right.\) \(A_{3}=\left\\{\text { first child has influenzal, } A_{4}=\\{ \text { second child has influ- }\right.\) enzal, \(B=\) lat least one child has influenzal, \(C=\) lat least one parent has influenzal, and \(D=\\{\) at least one person in the family has influenzal.What does \(A_{3} \cup B\) mean?

Piriformis syndrome is a pelvic condition that involves malfunction of the piriformis muscle (a deep buttock muscle), which often causes back and buttock pain with sciatica(pain radiating down the leg). An electrophysiologic test to detect piriformis syndrome involves measuring nerveconduction velocity (NCV) at two nerves in the leg (the tibial and peroneal nerves) with the leg flexed in a specific position. Increases in \(\mathrm{NCV}\) in these nerves are often associated with piriformis syndrome. The resulting test, called the flexion abduction and internal rotation (FAIR) test, is positive if the average \(\mathrm{NCV}\) in these nerves is delayed by \(2+\) seconds relative to normal.A small study compared the FAIR test results with patient self-reports of how they feel on a visual analog scale (VAS) of \(0-10,\) with 0 indicating no pain and 10 very severe pain. The results were as shown in Table 3.17 .Suppose physicians consider the FAIR test the gold standard, with a FAIR test result of \(\geq 2\) defined as a true positive and a FAlR test result of \(<2\) defined as a true negative. Suppose a VAS of \(\leq 4\) is considered a good clinical response based on self-report (a test-negative) and a VAS of \(\geq 5\) is considered a bad clinical response (a test-positive).What is the specificity of the VAS?

Consider a family with a mother, father, and two children. Let \(A_{1}=\left\\{\text { mother has influenzal, } A_{2}=\\{ \text { father has influenzal, }\right.\) \(A_{3}=\left\\{\text { first child has influenzal, } A_{4}=\\{ \text { second child has influ- }\right.\) enzal, \(B=\) lat least one child has influenzal, \(C=\) lat least one parent has influenzal, and \(D=\\{\) at least one person in the family has influenzal.Express \(D\) in terms of \(B\) and \(C .\)

Suppose an influenza epidemic strikes a city. In \(10 \%\) of families the mother has influenza; in \(10 \%\) of families the father has influenza; and in \(2 \%\) of families both the mother and father have influenza.Are the events \(A_{1}=\\{\text { mother has influenza }\\}\) and \(A_{2}=\) father has influenzal independent? Suppose there is a \(20 \%\) chance each child will get influenza, whereas in \(10 \%\) of two-child families both children get the disease.
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