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Chapter 13: Problem 56

A sample of \(n=500(x, y)\) pairs was collected and a test of \(H_{0}: \rho=0\) versus \(H_{a}: \rho \neq 0\) was carried out. The resulting \(P\) -value was computed to be \(.00032\). a. What conclusion would be appropriate at level of significance \(.001\) ? b. Does this small \(P\) -value indicate that there is a very strong linear relationship between \(x\) and \(y\) (a value of \(\rho\) that differs considerably from zero)? Explain.

Short Answer

Expert verified
a. The null hypothesis can be rejected at the .001 level of significance, thus implying there is a correlation. b. The small p-value is indicating that there is strong evidence against the null hypothesis, i.e., the correlation is not zero. However, the strength of the linear relationship would need to be gauged via the value of the correlation coefficient.

Step by step solution

01

Understand the Exercise

Before any computation, the question has to be understood. This is a hypothesis testing problem. The null hypothesis \(H_{0}: \rho=0\) is stating that there is no correlation. The alternative hypothesis \(H_{a}: \rho \neq 0\) is stating that there is a correlation. The p-value of .00032 has been computed from the data which represents the probability that the result from the sample was due to chance.
02

Interpret the P-Value

The computed p-value .00032, is then compared with the significance level, .001. A p-value less than or equal to the significance level (.001 in this case) means that the null hypothesis can be rejected. Since .00032 < .001, therefore the null hypothesis \(H_{0}: \rho=0\) can be rejected at the .001 significance level. This implies there is enough evidence to conclude that there is a correlation.
03

Meaning of Small P-Value

A small p-value indicates that there is strong evidence against the null hypothesis. In this case, that means there is strong evidence that the correlation between x and y (\(\rho\)) is not zero. However, this does not necessarily mean that the linear relationship between x and y is very strong, but simply that it is not zero. The actual value of the correlation coefficient would provide information on the strength and direction of the linear relationship.

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Key Concepts

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

p-value interpretation
In hypothesis testing, the p-value helps us determine the strength of evidence against the null hypothesis. It represents the probability of observing data as extreme as, or more extreme than, the results actually obtained, given that the null hypothesis is true.
  • If the p-value is low (typically less than or equal to a predetermined significance level), it suggests that the observed data is unlikely under the null hypothesis, leading to its rejection.
  • If the p-value is high, it implies insufficient evidence to reject the null hypothesis, indicating that the observed data is consistent with the null hypothesis.

In this particular problem, the p-value of .00032 is much lower than the significance level of .001, which suggests strong evidence against the null hypothesis.
null hypothesis
The null hypothesis is a statement used in statistics that proposes no significant effect or no relationship between variables in a study or experiment. In the context of this exercise, the null hypothesis is stated as:
  • \( H_{0} : \rho = 0 \)
This implies that there is no correlation between the variables \(x\) and \(y\).

It's important because it sets the baseline that the new data is tested against. By rejecting the null hypothesis, we conclude that there is enough statistical evidence to support some change or effect - in this case, a correlation between \(x\) and \(y\).
correlation coefficient
The correlation coefficient, represented as \( \rho \) (rho), measures the strength and direction of a linear relationship between two variables on a scatterplot. Possible values range from -1 to 1:
  • A coefficient of 1 indicates a perfect positive linear relationship.
  • A coefficient of -1 indicates a perfect negative linear relationship.
  • A coefficient of 0 indicates no linear relationship.

In this exercise, the test checks whether \( \rho eq 0 \), meaning that it is questioning whether any form of correlation exists between \( x \) and \( y \). Even if we reject the null hypothesis, it doesn't imply a strong linear correlation unless we know \( \rho \)'s magnitude.
significance level
The significance level, often denoted by \( \alpha \), is the criterion used for deciding whether to accept or reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true. Common levels are 0.05, 0.01, and 0.001.
In this exercise, the chosen significance level is 0.001. A rigorous threshold such as this indicates a desire for very strong evidence before rejecting the null hypothesis.
This level acts as a standard for comparison against the p-value. If the p-value is less than or equal to the significance level, there is sufficient evidence to reject the null hypothesis, signifying potential significance or impact of the observed data.

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