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Chapter 12: Problem 29

When testing the significance of the correlation coefficient, what is the alternative hypothesis?

Short Answer

Expert verified
The alternative hypothesis is \( H_1: \rho \neq 0 \), suggesting a significant correlation.

Step by step solution

01

Understand the Nature of Hypotheses in Correlation

In statistics, when testing hypotheses about a correlation coefficient, we usually have a null hypothesis and an alternative hypothesis. The null hypothesis asserts that there is no correlation between the two variables, meaning the population correlation coefficient, denoted by \( \rho \), is equal to zero (\( \rho = 0 \)).
02

Define the Alternative Hypothesis

The alternative hypothesis is designed to be in opposition to the null hypothesis. When we are testing for the significance of a correlation coefficient, we want to know if \( \rho \), the population correlation coefficient, is different from zero. Therefore, the alternative hypothesis states that there is a significant correlation between the two variables.
03

Formulate the Alternative Hypothesis

Based on the explanation in Step 2, the alternative hypothesis is formulated as \( H_1: \rho eq 0 \). This hypothesis suggests that the correlation coefficient is not zero, indicating that there is a significant relationship between the two variables.

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Key Concepts

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

Understanding the Null Hypothesis
In hypothesis testing for a correlation coefficient, the null hypothesis plays a crucial role. It helps determine if there is a meaningful relationship between two variables. Here's how it works:
  • The null hypothesis, often denoted as \( H_0 \), proposes that there is no correlation between the variables.
  • This means that the population correlation coefficient \( \rho \) is equal to zero, \( \rho = 0 \).
  • By stating "no correlation," we mean that any observed relationship in the sample data occurred by random chance alone.
When testing a correlation with a null hypothesis, our goal is usually to prove this statement wrong. If sufficient evidence is found against \( H_0 \), we might reject the hypothesis and explore other possibilities, like the alternative hypothesis.
Exploring the Alternative Hypothesis
When conducting statistical tests for correlations, the alternative hypothesis is what we consider if the null hypothesis is refuted. This represents the idea that there is a noticeable correlation between variables.
  • The alternative hypothesis is denoted as \( H_1 \).
  • It asserts that the population correlation coefficient \( \rho \) is not zero, \( H_1: \rho eq 0 \).
  • This points to a significant relationship between the variables, meaning any observed pattern or trend in the sample data is likely reflecting a real effect in the population.
By testing the alternative hypothesis, we aim to uncover new insights about how the variables interact. A significant result can drive further research and encourage deeper investigation into the variables involved.
Understanding Population Correlation Coefficient
The population correlation coefficient, represented by the symbol \( \rho \), is a key statistical measure that describes the strength and direction of a linear relationship between two variables at the population level.
  • \( \rho \) is often calculated from sample data and provides an estimate of the correlation for the entire population.
  • A \( \rho \) of 0 indicates no linear relationship, while a \( \rho \) close to 1 or -1 indicates a strong positive or negative relationship, respectively.
  • The significance testing of \( \rho \) enables researchers to determine if observed correlations in their sample are likely to exist in the population.
Understanding \( \rho \) is vital in many fields, as it helps quantify relationships between variables, thereby aiding in predictions, decision-making, and drawing meaningful conclusions from data.

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Most popular questions from this chapter

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where \(x\) is the number of endorsements the player has and \(y\) is the amount of money made (in millions of dollars). $$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\\ \hline\end{array}$$ Table 12.13 Draw a scatter plot of the data.

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where \(x\) is the number of endorsements the player has and \(y\) is the amount of money made (in millions of dollars). $$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\\ \hline\end{array}$$ Table 12.13 Use regression to find the equation for the line of best fit.

Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is \(y=12,000 x\) . What are the independent and dependent variables?

Ornithologists, scientists who study birds, tag sparrow hawks in 13 different colonies to study their population. They gather data for the percent of new sparrow hawks in each colony and the percent of those that have returned from migration. Percent return:74; 66; 81; 52; 73; 62; 52; 45; 62; 46; 60; 46; 38 Percent new:5; 6; 8; 11; 12; 15; 16; 17; 18; 18; 19; 20; 20 a. Enter the data into your calculator and make a scatter plot. b. Use your calculator鈥檚 regression function to find the equation of the least-squares regression line. Add this to your scatter plot from part a. c. Explain in words what the slope and y-intercept of the regression line tell us. d. How well does the regression line fit the data? Explain your response. e. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain. f. An ecologist wants to predict how many birds will join another colony of sparrow hawks to which 70% of the adults from the previous year have returned. What is the prediction?

Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,345 acres. The rate at which one person can mow is as follows: \(\hat{y}=1350-1.2 x\) where \(x\) is the number of hours and \(\hat{y}\) represents the number of acres left to mow. How many hours will it take to mow all of the lawns? (When is \(\hat{y}=0 ?\)
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