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Chapter 12: Problem 28

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

Short Answer

Expert verified
The null hypothesis is \( \rho = 0 \), indicating no linear relationship.

Step by step solution

01

Understanding the Concept of Correlation Coefficient

The correlation coefficient, often denoted by \( r \), is a measure of the strength and direction of a linear relationship between two variables. A common approach in statistics is to test the significance of this correlation to determine if it is different from zero, which would indicate no linear relationship.
02

Formulating the Null Hypothesis

The null hypothesis, denoted as \( H_0 \), is a statement that there is no effect or no difference. In the context of testing the significance of a correlation coefficient, the null hypothesis states that the population correlation coefficient \( \rho \) is equal to zero (\( \rho = 0 \)). This means that there is no linear relationship between the variables in the population.
03

Setting Up the Alternative Hypothesis

Though not strictly part of the question, for completeness, the alternative hypothesis should be mentioned. It can be directional (\( \rho > 0 \) or \( \rho < 0 \)) or non-directional (\( \rho eq 0 \)), suggesting that there is a correlation.
04

Implications of the Null Hypothesis

By testing the null hypothesis, we aim to determine if the observed correlation in our sample could be due to random chance alone. If we can reject the null hypothesis, it provides evidence that there is a statistically significant correlation in the population.

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Key Concepts

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

Understanding the Correlation Coefficient
The correlation coefficient, represented by the letter \( r \), is a fascinating statistical tool. Its main job is to quantify how two variables are related to each other. In simpler terms, it tells us if changes in one variable might be associated with changes in another. This coefficient ranges from -1 to 1.
  • If \( r = 1 \), there is a perfect positive linear relationship, meaning as one variable increases, so does the other.
  • Conversely, if \( r = -1 \), there is a perfect negative linear relationship, indicating that as one variable increases, the other decreases.
  • An \( r \) close to 0 means there is little to no linear relationship between the variables.
The strength and direction of the relationship are what make the correlation coefficient so crucial. It acts like a statistician's magnifying glass, providing a closer look at the potential connections between data sets.
Understanding Statistical Significance
When we test correlations, it's not just about finding some numerical value. We want to know if that value is meaningful. That's where statistical significance comes in. It helps us determine if the correlation we found in our data is likely to exist in the wider population or simply due to random chance.
In hypothesis testing:
  • The null hypothesis posits that there's no effect or relationship present (in correlation, \( \rho = 0 \)).
  • A statistically significant result allows us to reject this null hypothesis, suggesting the observed correlation is unlikely to be due to randomness.
  • Typically, a p-value is used to gauge this, with a common threshold of 0.05. If the p-value is below this, we declare our result significant and reject the null hypothesis.
Understanding this helps ensure that any conclusions are robust and reliable, not just flukes of random data.
Exploring the Alternative Hypothesis
While the null hypothesis states that there is no relationship, the alternative hypothesis gives us the possibility of an actual correlation. It's like the hypothesis version of a 'what if'.
For testing correlation coefficients, the alternative hypothesis can take several forms:
  • Non-directional: \( \rho eq 0 \). This suggests just any correlation, positive or negative.
  • Directional: \( \rho > 0 \) or \( \rho < 0 \), suggesting a specific direction of correlation, either positive or negative respectively.
Choosing which alternative hypothesis to test depends on prior knowledge or the theory being examined. It guides researchers on what potential outcomes would support their theories. By considering the alternative hypothesis, we acknowledge and explore the possibility that there is more than meets the eye in the data.

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Most popular questions from this chapter

Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is \(y=15-1.5 x\) where \(x\) is the number of hours passed in an eight-hour day of trading. What are the slope and y-intercept? Interpret their meaning.

Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up- front fee of \(\$ 25\) and another fee of \(\$ 12.50\) an hour. What are the dependent and independent variables?

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?

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 What does an \(r\) value of zero mean?

If the level of significance is 0.05 and the \(p\) -value is \(0.06,\) what conclusion can you draw?
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