91Ó°ÊÓ

A sample of \(n=500(x, y)\) pairs was collected and a test of \(H_{0}: \rho=0\) versus \(H_{\mathrm{a}}: \boldsymbol{\rho} \neq 0\) 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 0)? Explain. c. Now suppose a sample of \(n=10,000(x, y)\) pairs resulted in \(r=.022\). Test \(H_{0}: \rho=0\) versus \(H_{\mathrm{a}}: \rho \neq 0\) at level \(.05\). Is the result statistically significant? Comment on the practical significance of your analysis.

Step by step solution

01

Interpret the Given P-value for Part a

For part a, we should compare the given \(P\)-value, 0.00032, with the significance level \(\alpha = 0.001\). Since 0.00032 is less than 0.001, we reject the null hypothesis \(H_0: \rho = 0\).

02

Conclusion for Part a

Based on the comparison in Step 1, the conclusion is that there is sufficient evidence to suggest that there is a significant linear relationship between the variables \(x\) and \(y\) at the 0.001 significance level.

03

Examine the Strength of Relationship for Part b

For part b, the small \(P\)-value indicates that the relationship is statistically significant, but it does not measure the strength or magnitude of the relationship. A small \(P\)-value can occur even with a very small correlation coefficient when we have a large sample size. Hence, a very small \(P\)-value does not necessarily imply a strong linear relationship between \(x\) and \(y\).

04

Calculate the Test Statistic for Part c

For a sample size of \(n=10,000\) pairs with \(r=0.022\), we compute the test statistic using the formula: \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \]First calculate:\( t = \frac{0.022 \sqrt{9998}}{\sqrt{1-0.022^2}} \)\( t \approx \frac{0.022 \times 99.99}{1} \approx 2.2 \).

05

Look up Critical Value and Conclusion for Part c

With \(n = 10,000\), the degrees of freedom \(df = 9998\). At \(\alpha = 0.05\), the critical \(t\)-value for a two-tailed test is about 1.96. Since the computed \(t\)-value of 2.2 exceeds 1.96, we reject \(H_0: \rho = 0\), indicating the result is statistically significant.

06

Consider Practical Significance for Part c

The result is statistically significant, but the correlation coefficient \(r = 0.022\) is very small. This suggests that even though there is a statistically significant relationship, it may not be of practical significance, meaning the linear relationship is weak and unlikely to be meaningful in real-world applications.

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Key Concepts

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

Correlation Analysis

Correlation analysis is a technique used to measure the strength and direction of a relationship between two variables. In hypothesis testing, we often assess whether there is a significant correlation between the variables, typically denoted as \(\rho\) (rho). If \(\rho = 0\), it suggests no linear relationship exists.
This analysis is crucial because it helps us understand if changes in one variable might be related to changes in another, guiding decision-making and predictions.

• A positive correlation means as one variable increases, the other tends to increase.
• A negative correlation means as one variable increases, the other tends to decrease.
• No correlation implies no predictable relationship between the variables.

The correlation coefficient, \(r\), quantifies this relationship, ranging from -1 to 1. Values closer to 1 or -1 indicate a strong relationship, while values near 0 suggest a weak relationship.

Significance Level

The significance level, often denoted as \(\alpha\), is the threshold at which we decide whether to reject the null hypothesis. Common levels are \(0.05\) and \(0.001\).
In the exercise, when we have \(\alpha = 0.001\), it means we are only 0.1% willing to reject the null hypothesis by mistake (Type I error).
This makes it a stringent test for determining statistical significance.

• If the \(P\)-value is less than \(\alpha\), we reject the null hypothesis.
• If the \(P\)-value is greater than \(\alpha\), we fail to reject the null hypothesis.

This forms the crux of decision-making in hypothesis testing, aligning with how strict we want to be about avoiding incorrect conclusions.

P-value Interpretation

The $P$-value tells us the probability of observing our data if the null hypothesis is true. A small $P$-value suggests that such data is unlikely under the null hypothesis.
In our scenario, a $P$-value of $0.00032$ is substantially smaller than the significance level of $0.001$. This implies strong evidence against the null hypothesis, suggesting a significant relationship between $x$ and $y$.

• A $P$-value lower than 0.05 typically indicates statistical significance.
• The smaller the $P$-value, the stronger the evidence against the null hypothesis.

However, it is key to understand that a small $P$-value does not measure the effect's size, only its significance.

Statistical vs Practical Significance

Statistical significance signals when a result is unlikely to have occurred under the null hypothesis. Practical significance, however, considers whether the size of the effect is large enough to be meaningful in real-world contexts.
In the given exercise, the correlation coefficient $r = 0.022$ was statistically significant with a very large sample. Despite this, it is practically insignificant as the relationship is weak.

• Statistical significance can be influenced by large sample sizes, seemingly making negligible differences seem important.
• Practical significance is about the real-world applicability and impact of the findings.

Thus, it is essential to assess both elements to make informed decisions, particularly in applied fields where the outcome must have tangible relevance.