Question

Give the rejection region for a test to detect rank correlation if the number of pairs of ranks is 25 and you have these \(\alpha\) -values: a. \(\alpha=.05\) b. \(\alpha=.01\)

Short answer

Answer: For α = 0.05, the rejection region is r_s < -2.0687 or r_s > 2.0687. For α = 0.01, the rejection region is r_s < -2.8073 or r_s > 2.8073.

Step-by-step solution

Identify the distribution of the test statistic

In order to find the critical values, we need to know the distribution of the test statistic. For the Spearman's correlation coefficient test, the test statistic typically follows the T-distribution when the sample size is greater than 10. In this specific case, the number of pairs of ranks is 25, so the test statistic follows a T-distribution.

Find the degrees of freedom

To find the critical values, we need to determine the degrees of freedom associated with the T-distribution. For the Spearman's correlation coefficient test, the degrees of freedom are equal to the sample size (number of pairs of ranks) minus 2: $$df = n - 2$$For this exercise, the degrees of freedom will be: $$df = 25 - 2 = 23$$

Find the critical values for each α-value

Now that we have identified the distribution and degrees of freedom, we can find the critical values for each α-value. We will use a T-distribution table to look up these values for α = 0.05 and α = 0.01. a. For α = 0.05, and df=23, the critical values are: $$\pm 2.0687$$ b. For α = 0.01, and df=23, the critical values are: $$\pm 2.8073$$

Determine the rejection region for each α-value

Finally, we can state the rejection regions for the test based on the obtained critical values: a. For α = 0.05, the rejection region is: $$r_s < -2.0687 \ \text{or} \ r_s > 2.0687$$ b. For α = 0.01, the rejection region is: $$r_s < -2.8073 \ \text{or} \ r_s > 2.8073$$ In summary, if the calculated Spearman's rank correlation coefficient falls into the rejection region for a specific α-value, it indicates that there is significant evidence to reject the null hypothesis of no correlation between the two ranked variables with that specified significance level.

Key concepts

T-distribution

The T-distribution is essential in statistics, especially when dealing with small sample sizes. This type of distribution is symmetrical, bell-shaped, and similar to the standard normal distribution. However, it has heavier tails. This means it can better accommodate the variability found in smaller datasets.

The T-distribution becomes especially useful when the sample size is fewer than 30 and when the population standard deviation is unknown. Key features include:

• Symmetrical and bell-shaped
• Dependent on degrees of freedom
• As degrees of freedom increase, it approaches a normal distribution

In the context of Spearman's rank correlation, when the number of pairs (n) exceeds 10, the T-distribution provides a reliable framework to test hypotheses about ranks.

Degrees of Freedom

Degrees of freedom (often abbreviated as df) are vital in determining the correct distribution to use for statistical tests. In simple terms, degrees of freedom refer to the number of independent values we can freely vary in an analysis, without jeopardizing a constraint.

For the T-distribution in hypothesis testing, the degrees of freedom typically relate to the sample size. The formula for calculating it in Spearman's correlation is:\[ df = n - 2 \]where \( n \) is the number of pairs of ranks.

Thus, for our exercise with 25 pairs, the degrees of freedom are:\[ df = 25 - 2 = 23 \]Understanding degrees of freedom is vital because it affects the shape of the T-distribution, impacting the location of critical values.

Critical Values

Critical values are thresholds that help determine the significance of a test statistic. In hypothesis testing, they define boundaries where you can deduce if your results are statistically significant. We calculate them using a significance level (denoted by \( \alpha \)) and degrees of freedom.

For Spearman's test, we look at a T-distribution table to find critical values that correspond to both the chosen \( \alpha \) and df.

In the exercise, for \( \alpha = 0.05 \) and df = 23, the critical values were \( \pm 2.0687 \). Whereas, for \( \alpha = 0.01 \) the critical values were \( \pm 2.8073 \). These values are pivotal as they mark the boundaries within which we decide to reject or not reject the null hypothesis.

Rejection Region

The rejection region is the range of values where, if a test statistic falls within, it leads you to reject the null hypothesis. It is established using critical values obtained from your significance level and t-distribution.

For Spearman’s rank correlation, the rejection region is two-tailed, meaning that it considers extreme values on either side of the distribution.

• For \( \alpha = 0.05 \), the rejection region is below \(-2.0687\) or above \(2.0687\).
• For \( \alpha = 0.01 \), it expands to below \(-2.8073\) or above \(2.8073\).

This gives a clearer picture: If your test statistic falls in these regions, it implies strong evidence against the null hypothesis, concluding there is a significant correlation between the two sets of data.