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Chapter 7: Problem 27

The equation of the least-squares regression line for predicting the mean coral growth of a reef from the mean sea surface temperature is growth \(=6.98-0.22 \times\) sea surface temperature What does the slope of \(-0.22\) tell us? (a) The mean coral growth of reefs in the study is decreasing \(0.22\) centimeter per year. (b) The predicted mean coral growth of reefs in the study is \(0.22\) centimeter per degree of mean sea surface temperature. (c) The predicted mean coral growth of a reef in the study when the mean sea surface temperature is 0 is \(6.98\) centimeters.

Short Answer

Expert verified
(b) The predicted mean coral growth decreases by 0.22 cm per degree of temperature.

Step by step solution

01

Identify Regression Line Equation

The given equation for the regression line is \( \text{growth} = 6.98 - 0.22 \times \text{sea surface temperature} \). This represents a linear relationship between coral growth and sea surface temperature.
02

Understand the Components of the Equation

In the standard form \( y = mx + b \), \( m \) is the slope, and \( b \) is the y-intercept. Here, \( -0.22 \) is the slope \( m \), and \( 6.98 \) is the y-intercept \( b \).
03

Interpret the Slope

The slope \( -0.22 \) indicates that for each one-degree increase in sea surface temperature, the predicted mean coral growth decreases by 0.22 centimeters. This reflects the negative relationship between temperature and growth.
04

Match Explanation with Options

Compare the interpretation of the slope with the provided options: (a) refers to a yearly change, (b) refers to change per degree, and (c) talks about the intercept. The correct interpretation of the slope matches option (b).

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Key Concepts

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

Least-Squares Regression
Least-squares regression is a common technique used to find the best-fitting line through a set of data points. By minimizing the sum of the squares of the vertical distances of the points from the line, it ensures the line represents the trend of the data as closely as possible. The purpose of least-squares regression is to predict or estimate the dependent variable based on the independent variable.

For example, in the given regression equation for coral growth:
  • The dependent variable is the coral growth, which we aim to predict.
  • The independent variable is the sea surface temperature, which influences coral growth.
The formula for the best-fit line is often written as \( y = mx + b \), where \( y \) is the dependent variable, \( m \) is the slope, \( x \) is the independent variable, and \( b \) is the intercept. The slope \( m \) and intercept \( b \) are derived through the least-squares method.
Correlation
Correlation refers to a statistical measure that describes the extent to which two variables are linearly related. It's crucial to understand that correlation does not imply causation, but it indicates whether and how strongly pairs of variables are associated.

Correlation can be positive, negative, or zero:
  • A positive correlation means that as one variable increases, the other also increases.
  • A negative correlation means that as one variable increases, the other decreases.
  • A zero correlation implies no linear relationship between the variables.
In the context of our regression equation, the slope of \(-0.22\) suggests a negative correlation between the sea surface temperature and coral growth. As the temperature rises, the predicted coral growth declines.
Slope Interpretation
Interpreting the slope in a regression equation is key to understanding the relationship between the variables. The slope is the coefficient of the independent variable and indicates the rate of change in the dependent variable per unit change in the independent variable.

In the equation \( ext{growth} = 6.98 - 0.22 \times ext{sea surface temperature} \):
  • The slope is \(-0.22\). This means for each degree increase in sea surface temperature, there is an expected decrease of 0.22 centimeters in coral growth.
  • It indicates a negative relationship, where warmer sea temperatures can potentially affect coral growth negatively.
Understanding this helps researchers and environmentalists make predictions and decisions regarding coral reefs based on changing sea surface temperatures.

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Most popular questions from this chapter

A report says that "the median credit card debt of American households is zero." We know that many households have large amounts of credit card debt. In fact, the mean household credit card debt is close to \(\$ 8000\). Explain how the median debt can nonetheless be zero.

The heights of male five-year-olds have a Normal distribution with a mean of \(44.8\) inches and a standard deviation of \(2.1\) inches. \({ }^{6}\) (a) What percent of male five-year-olds have heights between 40 and 50 inches? (b) What range of heights covers the central \(95 \%\) of this distribution? (c) You are informed by your doctor that your five-year-old boy's height is at the 70th percentile of heights? How tall is your child?

What are the most important differences between female and male students? (a) A greater percentage of females spend five or more hours per day playing video or computer games or using the computer for something that is not school work, on an average school day, than males. (b) Females are more than twice as likely to spend no time playing video or computer games or using the computer for something that is not school work, on an average school day, than males. (c) A greater percentage of females spend one hour or less per day playing video or computer games or using the computer for something that is not school work, on an average school day, than males. (d) All of the above.

Python eggs. How is the hatching of water python eggs influenced by the temperature of the snake's nest? Researchers placed 104 newly laid eggs in a hot environment, 56 in a neutral environment, and 27 in a cold environment. Hot duplicates the warmth provided by the mother python. Neutral and cold are cooler, as when the mother is absent. The results: 75 of the hot eggs hatched, along with 38 of the neutral eggs and 16 of the cold eggs. 26 (a) Make a two-way table of "environment temperature" against "hatched or not." (b) The researchers anticipated that eggs would hatch less well at cooler temperatures. Do the data support that anticipation?

The number of adult Americans who smoke continues to drop. Here are estimates of the percents of adults (aged 18 and over) who were smokers in the years between 1965 and \(2014:^{15}\)-ll sMOKERS (a) Make a scatterplot of these data. (b) Describe the direction, fon, and strength of the relationship between percent of smokers and year. Are there any outliers? (c) Here are the means and standard deviations for both variables and the correlation between percent of smokers and year: $$ \begin{array}{rlr} \hat{x} & =1993.0, & s_{z}=14.3 \\ \hat{y} & =26.7, & s_{y}=7.0 \\ 99 r & =-0.99 & \end{array} $$ $$ x^{\circ}=1993.0,5 x=14.3 y^{\prime}=26.7,5 y=7.0 r=-0.99 r=-0.99 $$ Use this information to find the least-squares regression line for predicting percent of smokers from year and add the line to your plot. (d) According to your regression line, how much did smoking decline per year during this period, on the average? (e) What percent of the observed variation in percent of adults who smoke can be explained by linear change over time? (f) One of the govemment's national health objectives is to reduce smoking to no more than \(12 \%\) of adults by 2020 . Use your regression line to predict the percent of adults who will smoke in 2020 . (g) Use your regression line to predict the percent of adults who will smoke in \(2075 .\) Why is your result impossible? Why was it foolish to use the regression line for this prediction?
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