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91Ó°ÊÓ

Rank the correlations Consider each of the following relationships: the heights of fathers and the heights of their adult sons, the heights of husbands and the heights of their wives, and the heights of women at age 4 and their heights at age \(18 .\) Rank the correlations between these pairs of variables from highest to lowest. Explain your reasoning.

Short Answer

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1. Heights of women at age 4 and 18. 2. Heights of fathers and sons. 3. Heights of husbands and wives.

Step by step solution

01

Understanding Correlation

Before ranking the correlations, recall that a correlation measures the strength and direction of a linear relationship between two variables, ranging from -1 to 1.
02

Analyzing Father-Son Heights

The correlation between fathers' heights and their adult sons' heights tends to be relatively high due to genetic and environmental factors influencing height. This suggests a strong positive correlation.
03

Analyzing Husband-Wife Heights

The correlation between the heights of husbands and their wives is expected to be lower because height partner choice is influenced by social preferences rather than genetic factors. Thus, it's likely to be moderate.
04

Analyzing Child-Adult Woman Heights

The correlation between the heights of women at age 4 and their heights at age 18 will be high, as individual growth patterns are highly consistent over time, leading to a strong positive correlation.
05

Ranking the Correlations

Rank the correlations based on the analysis: highest to lowest. First, the correlation between girls' height at age 4 and 18, then father-son height correlation, and finally husband-wife height correlation.

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Key Concepts

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

Genetic Influence on Height
Genetic influence plays a crucial role in determining a person's height. It is estimated that approximately 60-80% of an individual's height is determined by genetic factors.
Family genes often dictate the general potential height that one may achieve. This is observed notably in the relationship between the heights of fathers and their adult sons.
Such a relationship shows a strong positive correlation primarily due to shared genetics. While genetics is a significant factor, it's important to note that other elements such as nutrition and overall health during the growth phase can also influence height. These environmental factors may slightly adjust the final outcome but are generally supportive rather than deterministic.
  • Genetic factors are predominant in influencing height.
  • Father-son height correlation is strong due to shared genes.
  • Environmental factors also play a supportive role.
Statistical Relationships
Understanding statistical relationships like correlation is essential when analyzing how variables relate to each other. A correlation is a statistical measure that expresses the extent to which two variables are linearly related.
It ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, -1 a perfect negative linear relationship, and 0 no linear relationship. In our analysis:
- The correlation between the heights of husbands and wives is expected to be lower. This is because the choice of a partner is influenced more by social preferences rather than genetics. Therefore, this relationship is often more moderate.
- The strong connection between the heights of women at age 4 and age 18 suggests a high correlation. Growth patterns, as we will explore further, seem consistent, thus showcasing this strong statistical bond.
  • Positive correlation indicates variables increase together.
  • The correlation range is between -1 (negative) to 1 (positive).
  • Moderate correlation seen in husband-wife height comparison.
Growth Patterns
Growth patterns describe the way in which children grow and develop over time. In particular, height at a young age can often predict future height, especially when growth is tracked over several years.
This is seen in the strong correlation between the height of women at age 4 and their height at age 18. Such patterns reflect biological development processes that generally follow a genetic blueprint.
Consistency in growth patterns means that even at a young age, height provides clues about adult stature. This is not only relevant for understanding personal growth but also for planning health interventions if necessary.
  • Predictable growth patterns indicate significant correlations at different ages.
  • Height at a young age can be an early indicator of adult height.
  • Growth patterns are typically genetically driven.

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Most popular questions from this chapter

The stock market Some people think that the behavior of the stock market in January predicts its behavior for the rest of the year. Take the explanatory variable \(x\) to be the percent change in a stock market index in January and the response variable \(y\) to be the change in the index for the entire year. We expect a positive correlation between \(x\) and \(y\) because the change during January contributes to the full year's change. Calculation from data for an 18 -year period gives $$ \begin{array}{c}{\overline{x}=1.75 \% \quad s_{x}=5.36 \% \quad \overline{y}=9.07 \%} \\ {s_{y}=15.35 \% \quad r=0.596}\end{array} $$ (a) Find the equation of the least-squares line for predicting full-year change from January change. Show your work. (b) The mean change in January is \(\overline{x}=1.75 \%\) . Use your regression line to predict the change in the index in a year in which the index rises 1.75\(\%\) in January. Why could you have given this result (up to roundoff error) without doing the calculation?

Smokers don't live as long (on average) as nonsmokers, and heavy smokers don't live as long as light smokers. You perform least-squares regression on the age at death of a group of male smokers \(y\) and the number of packs per day they smoked \(x .\) The slope of your regression line (a) will be greater than 0 . (b) will be less than \(0 .\) (c) will be equal to 0 . (d) You can't perform regression on these data. (e) You can't tell without seeing the data.

Husbands and wives The mean height of American women in their early twenties is 64.5 inches and the standard deviation is 2.5 inches. The mean height of men the same age is 68.5 inches, with standard deviation 2.7 inches. The correlation between the heights of husbands and wives is about \(r=0.5\) . (a) Find the equation of the least-squares regression line for predicting husband's height from wife's height. Show your work. (b) Use your regression line to predict the height of the husband of a woman who is 67 inches tall. Explain why you could have given this result without doing the calculation.

Acid rain Researchers studying acid rain measured the acidity of precipitation in a Colorado wilderness area for 150 consecutive weeks. Acidity is measured by pH. Lower pH values show higher acidity. The researchers observed a linear pattern over time. They reported that the regression line \(\mathrm{pH}=5.43-\) 0.0053 (weeks) fit the data well. (a) Identify the slope of the line and explain what itmeans in this setting. (b) Identify the \(y\) intercept of the line and explain what it means in this setting. (c) According to the regression line, what was the pH at the end of this study?

Gas mileage We expect a car's highway gas mileage to be related to its city gas mileage. Data for all 1198 vehicles in the government's 2008 Fuel Economy Guide give the regression line predicted highway mpg \(=4.62+1.109\) (city mpg). (a) What's the slope of this line? Interpret this value in context. (b) What's the intercept? Explain why the value of the intercept is not statistically meaningful. (c) Find the predicted highway mileage for a car that gets 16 miles per gallon in the city. Do the same for a car with city mileage 28 \(\mathrm{mpg.}\)

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