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Chapter 10: Problem 10

Over the past 30 years in the United States, there has been a strong negative correlation between the number of infant deaths at birth and the number of people over age \(65 .\) (a) Is the fact that people are living longer causing a decrease in infant mortalities at birth? (b) What lurking variables might be causing the increase in one or both of the variables? Explain.

Short Answer

Expert verified
No, longer life expectancy does not cause fewer infant deaths. Improvements in healthcare and socioeconomic conditions likely influence both trends.

Step by step solution

01

Understand Correlation vs. Causation

The problem presents a negative correlation between two variables: the number of infant deaths and the number of people over age 65. A correlation means that as one variable changes, the other tends to change in a consistent way, but it does not imply causation. To determine whether people living longer causes fewer infant mortalities, we must separate correlation from causation.
02

Evaluate Potential Causation

In (a), to answer if longer life expectancy among older adults causes a reduction in infant deaths, we must look for direct evidence or a mechanism that links these two. There is no direct biological or logical mechanism whereby older adults living longer reduces newborn deaths at birth; therefore, it is unlikely that the longer lifespan of older individuals directly causes a decrease in infant mortality.
03

Identify Lurking Variables

For (b), lurking variables are those that are not part of the study but affect the variables of interest. Possible lurking variables might include improvements in healthcare, economic conditions, nutrition, education levels, and public health initiatives. These factors could contribute to both increased lifespan and reduced infant mortality independently.
04

Analyze Healthcare Improvements

One major lurking variable is healthcare: advancements in medical technology, more accessible healthcare services, and better prenatal and elderly care can contribute to both trends. Better healthcare can reduce infant mortalities through improved prenatal and neonatal care and can help people live longer through advanced treatments and preventive care.
05

Consider Socioeconomic Factors

Socioeconomic improvements could also play a role. Increased wealth and education can lead to better overall health, evidenced by both higher average lifespans and improved infant survival rates. Thus, improved life quality could be a common factor influencing both variables.

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Key Concepts

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

Lurking Variables
When analyzing data, it is important to be aware of lurking variables. These are hidden factors that can influence the variables we are studying, despite not being directly measured. In the case of the negative correlation between infant mortality and an increasing elderly population, lurking variables might be responsible for the trends observed. For example, improvements in healthcare could enhance prenatal care, thus reducing infant mortality, while simultaneously improving elderly care, which extends life expectancy. Additionally, public health initiatives, such as better hygiene standards and vaccinations, might also play a role in both lowering infant deaths and increasing longevity for older adults. Recognizing these lurking variables helps prevent jumping to incorrect conclusions about cause-and-effect relationships in observed data trends.
Healthcare Advancements
Advancements in healthcare are often significant lurking variables in studies regarding population health changes. Over the years, medical technology has evolved dramatically, improving outcomes for both infants and the elderly.
  • In the neonatal arena, advancements include better prenatal monitoring, advanced delivery techniques, and improved neonatal intensive care units (NICUs), all contributing to decreased infant mortality.
  • For older adults, advancements in medical procedures, preventive medications, and chronic disease management have extended life expectancy.
This dual enhancement in care can contribute to seeming correlations in demographic data where both infant mortality decreases and elderly populations increase, although the exact interplay between these factors involves a combination of improvements rather than direct causation.
Socioeconomic Factors
Socioeconomic factors deeply influence health outcomes across all age groups. As societies become wealthier and more educated, general health improves significantly. This trend can be seen in the correlation between an increasing elderly population and decreasing infant mortality.
  • Wealthier societies can afford better healthcare infrastructure, offering more comprehensive services from prenatal to geriatric care.
  • Education influences health not only by increasing awareness of health practices but also by enabling access to better resources and information.
Thus, improvements in socioeconomic status tend to elevate public health, showcasing healthier, longer-lived populations alongside reduced rates of infant mortality. These factors serve as potential lurking variables that might explain trends in population health data without resorting to erroneous causal assumptions.

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Most popular questions from this chapter

Please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums \(\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma y^{2}\), and \(\sum x y\) and the value of the sample correlation coefficient \(\underline{r}\) (c) Find \(\bar{x}, \bar{y}, a\), and \(b .\) Then find the equation of the least- squares line \(\hat{y}=a+b x\) (d) Graph the least-squares line on your scatter diagram. Be sure to use the point \((\bar{x}, \bar{y})\) as one of the points on the line. (e) Find the value of the coefficient of determination \(r^{2} .\) What percentage of the variation in \(y\) can be explained by the corresponding variation in \(x\) and the least-squares line? What percentage is unexplained? Answers may vary slightly due to rounding. You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy some calves to add to the Bar-S herd. How much should a healthy calf weigh? Let \(x\) be the age of the calf (in weeks), and let \(y\) be the weight of the calf (in kilograms). The following information is based on data taken from The Merck Veterinary Manual (a reference used by many ranchers). $$ \begin{array}{r|rrrrrr} \hline x & 1 & 3 & 10 & 16 & 26 & 36 \\ \hline y & 42 & 50 & 75 & 100 & 150 & 200 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=92, \Sigma y=617, \Sigma x^{2}=2338, \Sigma y^{2}=\) \(82,389, \Sigma x y=13,642\), and \(r \approx 0.998 .\) (f) The calves you want to buy are 12 weeks old. What does the least- squares line predict for a healthy weight?

Over the past 50 years, there has been a strong negative correlation between average annual income and the record time to run 1 mile. In other words, average annual incomes have been rising while the record time to run 1 mile has been decreasing. (a) Do you think increasing incomes cause decreasing times to run the mile? Explain. (b) What lurking variables might be causing the increase in one or both of the variables? Explain.

Given the linear regression equation $$ x_{1}=1.6+3.5 x_{2}-7.9 x_{3}+2.0 x_{4} $$ (a) Which variable is the response variable? Which variables are the explanatory variables? (b) Which number is the constant term? List the coefficients with their corresponding explanatory variables. (c) If \(x_{2}=2, x_{3}=1\), and \(x_{4}=5\), what is the predicted value for \(x_{1}\) ? (d) Explain how each coefficient can be thought of as a "slope" under certain conditions. Suppose \(x_{3}\) and \(x_{4}\) were held at fixed but arbitrary values and \(x_{2}\) increased by 1 unit. What would be the corresponding change in \(x_{1} ?\) Suppose \(x_{2}\) increased by 2 units. What would be the expected change in \(x_{1}\) ? Suppose \(x_{2}\) decreased by 4 units. What would be the expected change in \(x_{1} ?\) (e) Suppose that \(n=12\) data points were used to construct the given regression equation and that the standard error for the coefficient of \(x_{2}\) is \(0.419\). Construct a \(90 \%\) confidence interval for the coefficient of \(x_{2}\). (f) Using the information of part (e) and level of significance \(5 \%\), test the claim that the coefficient of \(x_{2}\) is different from zero. Explain how the conclusion of this test would affect the regression equation.

Suppose two variables are positively correlated. Does the response variable increase or decrease as the explanatory variable increases?

The following data are based on information from the Harvard Business Review (Vol. 72, No. 1). Let \(x\) be the number of different research programs, and let \(y\) be the mean number of patents per program. As in any business, a company can spread itself too thin. For example, too many research programs might lead to a decline in overall research productivity. The following data are for a collection of pharmaceutical companies and their research programs: $$ \begin{array}{c|rrrrrr} \hline x & 10 & 12 & 14 & 16 & 18 & 20 \\ \hline y & 1.8 & 1.7 & 1.5 & 1.4 & 1.0 & 0.7 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=90, \Sigma y=8.1, \Sigma x^{2}=1420\), \(\Sigma y^{2}=11.83, \Sigma x y=113.8\), and \(r \approx-0.973 .\) (f) Suppose a pharmaceutical company has 15 different research programs. What does the least-squares equation forecast for \(y=\) mean number of patents per program?
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