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Chapter 5: Problem 15

What would be the correlation between the ages of husbands and wives if men always married woman who were (a) 3 years younger than themselves? (b) 2 years older than themselves? (c) half as old as themselves?

Short Answer

Expert verified
(a) +1, (b) +1, (c) +1, assuming the relationship maintains linearity.

Step by step solution

01

Understanding Correlation

First, we need to recall that correlation measures the strength and direction of a linear relationship between two variables. A perfect positive correlation, where two variables increase together, has a value of +1. A perfect negative correlation, where one variable decreases as the other increases, has a value of -1.
02

Calculating Correlation for Part (a)

In scenario (a), husbands always marry women who are 3 years younger than themselves. This suggests that for every age increase in the husband, the wife's age increases linearly by the same amount, maintaining a constant difference of 3 years. Thus, the relationship between their ages is perfectly linear and the correlation is \[ r = +1 \]
03

Calculating Correlation for Part (b)

For scenario (b), husbands marry women who are 2 years older. Similarly to (a), the relationship is perfectly linear despite the age difference now being negative for the husbands. An increase in the husband's age corresponds to a proportionate increase in the wife's age, maintaining a constant difference of -2 years. Hence, the correlation is also \[ r = +1 \]
04

Calculating Correlation for Part (c)

Scenario (c) involves husbands marrying women who are half as old as they are. This scenario introduces a non-linear relationship (as the age of the wife is a fraction of the husband's age), which means that the correlation is not a perfect linear relationship between the ages. However, if all husbands have the same proportion, correlation still holds, assuming ages don't vary widely, meaning that\[ r = +1 \] even in perception but will require more understanding in a variational context.

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Key Concepts

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

Linear Relationship
A linear relationship is a straight-line connection between two variables where the change in one variable is proportional to the change in another. This means if you have one variable increasing, the other will increase at a constant rate, which results in a straight line when plotted. For example, if you think about the original exercise where husbands marry wives of a constant age difference, that's a perfect display of linearity.
  • Concept of Proportional Change: Linear relationships are all about constant proportional changes. If a husband's age changes by a certain amount, the wife's age changes by exactly the same amount if there is a linear relationship of constant age difference as given in the example.
  • Graphical Representation: When illustrating this relationship on a graph with age of the husband on x-axis and age of the wife on y-axis, you'll see a perfectly straight line.
  • Positive or Negative: A linear relationship can be either positively or negatively sloped depending on whether the variables increase or decrease together.
Positive Correlation
Positive correlation occurs when two variables move in the same direction. If one increases, so does the other, and if one decreases, so does the other. In the exercise's context, the ages of husbands and wives showcase this perfectly because as the husbands' ages increase, so do the wives', despite a fixed age difference.
  • Value of +1: A positive correlation has a value of up to +1, indicating a perfect linear relationship where increases in husbands' ages correspond to increases in wives' ages.
  • Interpreting +1 Value: In scenarios a and b from the exercise, whether the difference is +3 or -2 in years, both illustrate positive correlations because the increase or decrease is consistent and proportional.
  • Practical Implication: This kind of correlation helps in forecasting and predictions. Knowing the trend helps to predict the wife's age if the husband's age is known because of their perfect correlation.
Age Difference
The age difference, in a correlation context, refers to the difference in years between two correlating ages, such as those of a husband and wife. This difference can affect how linear the relationship is perceived, but in the scenarios provided, it remains constant.
  • Meta Stability: Despite different age gaps (e.g., -3, +2 years), meta-stability in linear relationship keeps correlations positive.
  • Invariability: If variability in age difference doesn't occur, a constant age gap contributes to the perfect linear correlation.
  • Non-linear Perception: In scenario c, with the wife being half as old as the husband, the perceived relationship begins to slip from linear, yet calculated correlation can still suggest a degree of linear alignment due to maintained proportionality among similar cohorts.

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Most popular questions from this chapter

A study conducted at the University of Denver investigated whether babies take longer to learn to crawl in cold months, when they are often bundled in clothes that restrict their movement, than in warmer months. \(^{17}\) Infants born during the study year were split into twelve groups, one for each birth month. We consider the average crawling age of babies in each group against the average temperature when the babies are six months old (that's when babies often begin trying to crawl). Temperature is measured in degrees Fahrenheit ( \({ }^{\circ} \mathrm{F}\) ) and age is measured in weeks. (a) Describe the relationship between temperature and crawling age. (b) How would the relationship change if temperature was measured in degrees Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and age was measured in months? (c) The correlation between temperature in \({ }^{\circ} \mathrm{F}\) and age in weeks was \(R=-0.70 .\) If we converted the temperature to \({ }^{\circ} \mathrm{C}\) and age to months, what would the correlation be?

Is the gestational age (time between conception and birth) of a low birth- weight baby useful in predicting head circumference at birth? Twenty-five low birth-weight babies were studied at a Harvard teaching hospital; the investigators calculated the regression of head circumference (measured in centimeters) against gestational age (measured in weeks). The estimated regression line is head_circumference \(=3.91+0.78 \times\) gestational_age (a) What is the predicted head circumference for a baby whose gestational age is 28 weeks? (b) The standard error for the coefficient of gestational age is \(0.35,\) which is associated with \(d f=23\). Does the model provide strong evidence that gestational age is significantly associated with head circumference?

The scatterplot shows the relationship between socioeconomic status measured as the percentage of children in a neighborhood receiving reduced-fee lunches at school (lunch) and the percentage of bike riders in the neighborhood wearing helmets (helmet). The average percentage of children receiving reduced-fee lunches is \(30.8 \%\) with a standard deviation of \(26.7 \%\) and the average percentage of bike riders wearing helmets is \(38.8 \%\) with a standard deviation of \(16.9 \%\). (a) If the \(R^{2}\) for the least-squares regression line for these data is \(72 \%\), what is the correlation between lunch and helmet? (b) Calculate the slope and intercept for the leastsquares regression line for these data. (c) Interpret the intercept of the least-squares regression line in the context of the application. (d) Interpret the slope of the least-squares regression line in the context of the application. (e) What would the value of the residual be for a neighborhood where \(40 \%\) of the children receive reduced-fee lunches and \(40 \%\) of the bike riders wear helmets? Interpret the meaning of this residual in the context of the application.

Exercise 5.12 introduces data on the average monthly temperature during the month babies first try to crawl (about 6 months after birth) and the average first crawling age for babies born in a given month. A scatterplot of these two variables reveals a potential outlying month when the average temperature is about \(53^{\circ} \mathrm{F}\) and average crawling age is about 28.5 weeks. Does this point have high leverage? Is it an influential point?

Exercise 5.11 introduces data on the Coast Starlight Amtrak train that runs from Seattle to Los Angeles. The mean travel time from one stop to the next on the Coast Starlight is 129 mins, with a standard deviation of 113 minutes. The mean distance traveled from one stop to the next is 107 miles with a standard deviation of 99 miles. The correlation between travel time and distance is 0.636 . (a) Write the equation of the regression line for predicting travel time. (b) Interpret the slope and the intercept in this context. (c) Calculate \(R^{2}\) of the regression line for predicting travel time from distance traveled for the Coast Starlight, and interpret \(R^{2}\) in the context of the application. (d) The distance between Santa Barbara and Los Angeles is 103 miles. Use the model to estimate the time it takes for the Starlight to travel between these two cities. (e) It actually takes the the Coast Starlight about 168 mins to travel from Santa Barbara to Los Angeles. Calculate the residual and explain the meaning of this residual value. (f) Suppose Amtrak is considering adding a stop to the Coast Starlight 500 miles away from Los Angeles. Would it be appropriate to use this linear model to predict the travel time from Los Angeles to this point?
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