Question

For women, pregnancy lasts about 9 months. In other species of animals, the length of time from conception to birth varies. Is there any evidence that the gestation period is related to the animal's lifespan? The first scatterplot shows Gestation Period (in days) vs. Life Expectancy (in years) for 18 species of mammals. The highlighted point at the far right represents humans. a) For these data, \(r=0.54\), not a very strong relationship. Do you think the association would be stronger or weaker if humans were removed? Explain. b) Is there reasonable justification for removing humans from the data set? Explain. c) Here are the scatterplot and regression analysis for the 17 nonhuman species. Comment on the strength of the association. d) Interpret the slope of the line. e) Some species of monkeys have a life expectancy of about 20 years. Estimate the expected gestation period of one of these monkeys.

Short answer

Removing humans could strengthen the relationship; it's reasonable since humans are outliers. Without humans, the association appears stronger. The slope indicates gestation change per year of life expectancy. Use regression line to estimate for monkeys.

Step-by-step solution

Analyzing the Effect of Removing Humans

Since humans are the outlier with a large life expectancy compared to other mammals, removing them could potentially strengthen the correlation (increase the value of \(r\)), as they may skew the data. Therefore, without the human data point, the relationship might appear stronger.

Justification for Removing Humans

Humans have significantly different life expectancy and cultural factors influencing gestation that do not apply to other species. Therefore, it is reasonable to remove them to assess the typical relationship for non-human mammals.

Comment on Association Strength Without Humans

After removing humans, if the scatterplot and regression analysis show a correlation coefficient, \(r\), larger than 0.54, it indicates a stronger relationship. The analysis without humans likely shows a clearer trend if the outlier is removed.

Interpreting the Slope of the Regression Line

The slope of the line in a linear regression gives the expected change in the gestation period (in days) for each additional year of life expectancy. Thus, it can be interpreted as how much longer the gestation period tends to be for species that live one additional year.

Estimating Gestation Period for Monkeys

To estimate the gestation period for monkeys with a life expectancy of 20 years, use the linear equation obtained from the regression analysis: \( \text{Gestation Period} = \text{Intercept} + (20 \times \text{Slope}) \). Substituting 20 for life expectancy provides the estimated gestation period for these monkeys.

Key concepts

Scatterplot

A scatterplot is a type of graph that uses Cartesian coordinates to display values of two variables for a set of data. Each dot on the plot represents an observation. In the context of comparing gestation periods and life expectancy across different mammalian species, a scatterplot can be a valuable visualization tool.

The scatterplot reveals how gestation periods relate to life expectancy by displaying these two variables on the x and y axes. Clusters of points and their spread can indicate trends, patterns, or potential correlations. For instance, if points tend to form a line or curve, there might be a relationship between the variables. The scatterplot showing data for 18 species, including humans, can illustrate the degree and nature of the association between gestation period and lifespan.

• Helps in identifying outliers such as humans with a significantly different life expectancy.
• Useful for visually assessing if there is a linear relationship between the two variables.
• Provides a foundation to further analyze with other statistical tools like correlation and regression.

Correlation Coefficient

The correlation coefficient, often represented by the symbol \( r \), measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. A value close to 1 indicates a strong positive relationship, while a value close to -1 indicates a strong negative relationship. A value around 0 suggests no linear relationship.

In our exercise, the correlation coefficient \( r = 0.54 \) suggests a moderate positive relationship between gestation periods and life expectancy among the surveyed species.

Removing an outlier, such as the human data point, may increase \( r \) if humans skew the overall pattern observed among other mammals. This is because outliers can disproportionately influence the correlation, either masking or exaggerating the true relationship of the bulk of the data.

• Indicates the degree and direction of association between gestation and lifespan.
• Helps to gauge whether modifications, like removing outliers, make patterns clearer.
• Aids in determining how linear the observed relationship is.

Regression Analysis

Regression analysis examines the relationship between dependent and independent variables, often to make predictions or understand underlying patterns. It provides a line of best fit through data points in a scatterplot, allowing us to quantify the relationship.

The equation of the line, usually written as \( \text{Gestation Period} = \text{Intercept} + (\text{Slope} \times \text{Life Expectancy}) \), captures this relationship. Here, the slope represents the expected change in the gestation period for each additional year of life expectancy. This interpretation is crucial for understanding how one variable impacts the other in a predictable pattern.

For example, if the regression line after removing human data shows a slope that indicates increasing gestation days with more life expectancy years, this suggests a clearer pattern than when humans were included. For species like monkeys with a life expectancy of 20 years, we can substitute this value into the regression equation to estimate their gestation period.

• Used for predicting values, such as estimating a monkey's gestation period based on life expectancy.
• Helps in understanding the nature of the relationship between gestation and lifespan.
• Provides a statistical method to interpret and quantify patterns observed in a scatterplot.