Question

When \(n \geq 30,\) the formula \(r=\frac{\pm z}{\sqrt{n-1}}\) can be used to find the critical values for the rank correlation coefficient. For example, if \(n=40\) and \(\alpha=0.05\) for a two-tailed test, $$ r=\frac{\pm 1.96}{\sqrt{40-1}}=\pm 0.314 $$ Hence, any \(r_{s}\) greater than or equal to +0.314 or less than or equal to -0.314 is significant. Find the critical \(r\) value for each (assume that the test is two-tailed). $$ n=60, \alpha=0.10 $$

Short answer

The critical \(r\) value is approximately \(\pm 0.214\).

Step-by-step solution

Identify the Parameters

We begin with the parameters provided in the exercise. We are given \(n = 60\) and \(\alpha = 0.10\) for a two-tailed test. We will use these parameters to find the critical \(r\) value.

Determine the Z-Score for Alpha

Since \(\alpha = 0.10\) and it is a two-tailed test, we need to find the Z-score that corresponds to the cumulative probability of \(1 - \frac{\alpha}{2}\). This results in \(1 - \frac{0.10}{2} = 0.95\). From the standard normal distribution table, the Z-score for 0.95 is approximately \(1.645\).

Adjust the Z-Score by Sample Size

We use the formula for the critical value: \(r = \frac{\pm z}{\sqrt{n-1}}\). Substitute the known values: \(n = 60\) and \(z = 1.645\). The equation becomes \(r = \frac{\pm 1.645}{\sqrt{60 - 1}}\).

Calculate the Critical r Value

Calculate the denominator \(\sqrt{60 - 1}\) which simplifies to \(\sqrt{59}\). Calculate \(r = \frac{1.645}{\sqrt{59}}\). Using a calculator, \(\sqrt{59} \approx 7.68\). Then \(r = \frac{1.645}{7.68} \approx 0.214\). So, the critical \(r\) is approximately \(\pm 0.214\).

Key concepts

Understanding Two-Tailed Tests

A two-tailed test is a statistical method used to determine if there is a significant difference from a specific null hypothesis in either direction. Unlike a one-tailed test, which only checks for an increase or decrease, a two-tailed test considers both possibilities simultaneously. This is important for the rank correlation coefficient, as it allows us to see if the correlation is significantly different from zero, either positively or negatively.

When conducting a two-tailed test, you split the significance level (alpha) between the two tails of the distribution. This means that if your alpha level is 0.10, you assign 0.05 to the lower tail and 0.05 to the upper tail. This balanced approach helps ensure a more comprehensive understanding of deviation from the null hypothesis.

Role of Critical Values

Critical values are pivotal in hypothesis testing, serving as the threshold that determines whether a test statistic is significant. For rank correlation coefficients, critical values help us decide if the correlation observed is far enough from zero to rule out the possibility that it occurred by random chance.

The critical value is calculated using the Z-score and the formula given in the problem, taking into account the sample size (n). Once the critical values are determined, any test statistic that falls beyond these values implies significance, prompting a reject of the null hypothesis and suggesting a true correlation exists in the data.

Deciphering the Z-score

The Z-score is a key statistic that reflects how many standard deviations an element is from the mean. In the context of hypothesis testing, it helps us understand where a test statistic falls within the normal distribution.

To find the Z-score related to a specific alpha level for a two-tailed test, you calculate the cumulative probability equivalent to half the alpha level. For example, with an alpha of 0.10, you calculate for 0.05 in each tail, leading to finding the Z-score for a cumulative probability of 1 - 0.05 = 0.95. This approach helps in pinpointing the critical values needed to assess the statistical significance of the correlation coefficient.

Understanding the Alpha Level

The alpha level in hypothesis testing represents the probability threshold for rejecting the null hypothesis. Commonly set at 0.05 or 0.10, the alpha level indicates the risk you are willing to take in concluding a relationship exists when it does not.

In a two-tailed test, the total alpha level is divided between the two tails of the distribution. Thus, for an alpha of 0.10, each tail gets an alpha of 0.05. The Z-scores corresponding to this adjusted probability aid in defining the range of critical values.

• Lower alpha levels imply stricter tests, reducing the chances of committing a Type I error – falsely claiming significance.
• Understanding and setting this level is crucial, as it influences the sensitivity and specificity of your hypothesis testing.