91Ó°ÊÓ

A sample of \(n=500(x, y)\) pairs was collected and a test of \(H_{0}: \rho=0\) versus \(H_{\mathrm{a}}: \rho \neq 0\) 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 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

Analyze the P-value for significance level 0.001

The P-value is 0.00032, which is less than the significance level of 0.001. Therefore, we reject the null hypothesis \(H_0: \rho = 0\) in favor of the alternative hypothesis \(H_a: \rho eq 0\). Thus, it is statistically significant.

02

Interpret the small P-value

Although the P-value is very small, indicating statistical significance, it does not necessarily imply a very strong linear relationship between \(x\) and \(y\). The P-value merely indicates that \(\rho\) is different from zero, but it does not describe the magnitude of \(\rho\). A small \(P\)-value can occur even if \(\rho\) is very close to zero but is still significantly different from zero.

03

Compute the test statistic for n=10,000 and r=0.022

For the larger sample of \(n=10,000\) pairs, use the formula for the test statistic for Pearson correlation: \[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \]Substitute \(r=0.022\) and \(n=10,000\), \[ t = \frac{0.022 \times \sqrt{9998}}{\sqrt{1-0.022^2}} \approx \frac{0.022 \times 99.99}{0.999757} \approx 2.199 \]Using a significance level of \(\alpha=0.05\) and a large \(n\), compare \(t\) with \(z_{0.025}\approx1.96\). Since \(2.199 > 1.96\), the result is statistically significant, meaning \(\rho\) is different from zero.

04

Assess practical significance in Step 3

While the test indicates statistical significance, the correlation coefficient \(r = 0.022\) is very small. This suggests a very weak linear relationship, meaning the practical significance of the correlation is minimal despite the statistical significance.

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Key Concepts

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

P-value interpretation

The concept of the P-value can be a bit puzzling at first. In simple terms, a P-value helps you decide whether a result is statistically significant.

• When you perform a hypothesis test, you are trying to find out whether the observed data fits the model of having no effect (null hypothesis) or suggests an effect (alternative hypothesis).
• The P-value represents the probability of observing data as extreme as what was observed, assuming the null hypothesis is true.

In our exercise, a P-value of .00032 was found. This means that there's a very small chance that the observed correlation could have happened if the real correlation was zero. Since this P-value is less than the significance level of 0.001, we reject the null hypothesis and accept that there is statistically significant evidence to believe the correlation between x and y is not zero.

Pearson correlation

Pearson correlation is a statistical measure that expresses the extent of a linear relationship between two variables. It is denoted by the symbol \( r \).

• A Pearson correlation of +1 means that the two variables are perfectly positively linearly related.
• A correlation of -1 indicates a perfect negative linear relationship.
• A correlation of 0 means no linear relationship exists between the variables.

In our exercise, when interpreting the P-value, we determined that the correlation is not zero. However, it does not tell us how large the correlation is. Thus, while the P-value helped us reject the null hypothesis, only the correlation coefficient gives insight into the strength and direction of the relationship.

Statistical significance

Statistical significance is a mathematical measure of certainty that a relationship between two or more variables is caused by something other than chance.

• In hypothesis testing, if a test result is statistically significant at a given significance level (e.g., 0.05 or 0.001), it means there is strong evidence against the null hypothesis.
• For example, in the context of the exercise, the P-value of 0.00032 was considered statistically significant because it was less than 0.001.

Statistical significance indicates that the observed relationship is likely not due to random chance. However, it's important to remember that it doesn’t comment on the size or importance of the correlation.

Practical significance

While statistical significance tells us about the likelihood of an effect being real, practical significance considers whether the effect has real-world implications.

• An effect can be statistically significant but practically insignificant if the impact is too small to be meaningful in real life.
• In our example, the correlation coefficient \( r = 0.022 \) is statistically significant but indicates a very weak relationship.

This means that although we can say with confidence there's a linear relationship between \( x \) and \( y \), its actual impact might not be noticeable or useful in practical scenarios. Hence, always consider both statistical and practical significance!

Sample size effect

The size of a sample can significantly affect the results of hypothesis testing. Larger samples provide more reliable results but can also make small effects statistically significant.

• As sample size increases, the standard error decreases, often making it easier to detect small effects.
• This can lead to statistically significant results even if the actual effect size is trivial, as seen in our exercise with 10,000 pairs.

A large sample size can detect even small discrepancies from the null hypothesis. However, it's crucial to interpret these findings with the practical implications in mind. A very tiny effect might be statistically noticeable but practically meaningless. Balancing the interpretation of statistical tests with their real-world importance is key!