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Chapter 13: Problem 57

A sample of \(n=10,000(x, y)\) pairs resulted in \(r=\) .022. Test \(H_{0}: \rho=0\) versus \(H_{a}: \rho \neq 0\) at significance level .05. Is the result statistically significant? Comment on the practical significance of your analysis.

Short Answer

Expert verified
To arrive at a short answer, the value of Z from the calculation in Step 1 is needed for comparison with the critical value. If |Z| > 1.96, the result is statistically significant and reject the null hypothesis. If |Z| 鈮� 1.96, do not reject the null hypothesis. In terms of practical significance, considering the weak correlation of 0.022, there may not be any practical significance even if the result is statistically significant.

Step by step solution

01

Calculate the test statistic

The test statistic Z for the correlation coefficient is determined by the formula \(Z = r \sqrt{(n - 2) / (1 - r^2)}\). Substituting the given values, we have \(Z = 0.022 \sqrt{(10,000 - 2) / (1 - 0.022^2)}\). Compute this expression to obtain the numerical value of Z.
02

Determine the critical value

The problem gives a significance level of 0.05. Since the problem does not specify the type of test (one-tailed or two-tailed), we assume a two-tailed test because the alternative hypothesis \(H_{a}: \rho \neq 0\) doesn't specify a direction. In this case, we will reject the null hypothesis if our test statistic falls in the top 2.5% or bottom 2.5% of the standard normal distribution. That鈥檚 because under the null hypothesis, we assume the correlation coefficient will follow a standard normal distribution. The critical value for a two-tailed test at 0.05 level is approximately 1.96.
03

Make a decision

After obtaining the value of Z from Step 1, compare this with the critical value from Step 2. If the absolute value of Z is greater than the critical value, reject the null hypothesis in favor of the alternative hypothesis. Otherwise, do not reject the null hypothesis.
04

Consider practical significance

If the result is statistically significant, discuss its practical implications. Keep in mind that even though a result might be statistically significant, it might not have practical significance, especially if the correlation is very weak. In practical terms, a correlation of 0.022 is relatively weak, which means the variables are barely if at all, related.

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Key Concepts

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

Correlation Coefficient
The correlation coefficient, often represented by the symbol \( r \), is a statistical measure that describes the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where:
  • -1 indicates a perfect negative linear relationship,
  • 0 indicates no linear relationship,
  • 1 indicates a perfect positive linear relationship.
In this exercise, the given correlation coefficient is 0.022. This value is very close to 0, suggesting a very weak relationship between the variables.

It's important to remember that while the correlation coefficient measures the strength of a linear relationship, it doesn't imply causation. Even if the correlation is strong, it doesn't mean one variable causes the other to change.
Significance Level
The significance level, denoted by \( \alpha \), is a threshold used in hypothesis testing to decide whether an observed effect is statistically significant. It's usually set at 0.05, which means there's a 5% risk of concluding that a difference exists when there isn't one.

In our problem, the significance level is set at 0.05, which implies that the confidence level is 95%. This means we are 95% confident that our results are not due to random chance.

For a two-tailed test, like in our exercise, the critical region is split between both tails of the distribution curve. Therefore, the extreme values, both positive and negative, each cover 2.5% of the area under the curve, leading to critical values of approximately 卤1.96. If the test statistic falls beyond these values, the null hypothesis will be rejected.
Null Hypothesis
The null hypothesis, represented as \( H_0 \), is a statement used in hypothesis testing that proposes no effect or no difference. In the context of our exercise, it's defining that there is no correlation between the two variables (\( \rho = 0 \)).

The alternative hypothesis, \( H_a \), suggests that there is a correlation (\( \rho eq 0 \)).

The purpose of hypothesis testing is to determine whether there is enough evidence to reject the null hypothesis. In this situation, after computing the test statistic \( Z \), we would compare it against critical values. If |Z| is greater than the critical value, the null hypothesis is rejected, indicating that there is a statistically significant relationship between the variables. In contrast, if |Z| is less than 1.96, the null hypothesis fails to be rejected, meaning the evidence is not strong enough to suggest a correlation in the population.

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Most popular questions from this chapter

If the sample correlation coefficient is equal to 1 , is it necessarily true that \(\rho=1 ?\) If \(\rho=1\), is it necessarily true that \(r=1 ?\)

Suppose that a regression data set is given and you are asked to obtain a confidence interval. How would you tell from the phrasing of the request whether the interval is for \(\beta\) or for \(\alpha+\beta x^{*} ?\)

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Let \(x\) be the size of a house (sq \(\mathrm{ft}\) ) and \(y\) be the amount of natural gas used (therms) during a specified period. Suppose that for a particular community, \(x\) and \(y\) are related according to the simple linear regression model with \(\beta=\) slope of population regression line \(=.017\) \(\alpha=y\) intercept of population regression line \(=-5.0\) a. What is the equation of the population regression line? b. Graph the population regression line by first finding the point on the line corresponding to \(x=1000\) and then the point corresponding to \(x=2000\), and drawing a line through these points. c. What is the mean value of gas usage for houses with 2100 sq \(\mathrm{ft}\) of space? d. What is the average change in usage associated with a 1-sq-ft increase in size? e. What is the average change in usage associated with a 100-sq-ft increase in size? f. Would you use the model to predict mean usage for a 500-sq-ft house? Why or why not? (Note: There are no small houses in the community in which this model is valid.)
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