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). $$ n=50, \alpha=0.05 $$

Short answer

The critical \(r\) value for \(n=50\) and \(\alpha=0.05\) is \(\pm 0.28\).

Step-by-step solution

Identify Variables

First, identify the given variables from the problem. You have \(n = 50\) and \(\alpha = 0.05\) for a two-tailed test. The variable \(z\) corresponds to a critical value from the normal distribution, which for \(\alpha = 0.05\) in a two-tailed test is \(1.96\).

Calculate Square Root of (n - 1)

Next, calculate the square root of \(n - 1\). In this case, \(n = 50\), so \(n - 1 = 49\). Calculate \(\sqrt{49}\):\[\sqrt{49} = 7\]

Calculate r using Formula

Use the formula \(r = \frac{\pm z}{\sqrt{n - 1}}\). Substitute the values of \(z\) and \(\sqrt{n - 1}\) into the formula:\[r = \frac{\pm 1.96}{7}\]

Simplify the Expression

Simplify the expression to find the critical \(r\) value:\[r = \pm \frac{1.96}{7} = \pm 0.28\]

Interpret the Result

The critical values for the rank correlation coefficient \(r\) are \(+0.28\) and \(-0.28\). This means if a calculated rank correlation coefficient \(r_s\) for \(n = 50\) is greater than or equal to \(+0.28\) or less than or equal to \(-0.28\), it is considered significant for \(\alpha = 0.05\).

Key concepts

Critical Values

In the context of statistical tests, critical values are the threshold values that determine whether the result of a test is statistically significant. In the example provided, the critical value is used to determine significance in a hypothesis test involving a rank correlation coefficient, specifically Spearman's rank correlation coefficient. For problems involving rank correlation, when samples are large (i.e., when the sample size \(n\) is 30 or more), we often use the normal distribution to approximate behavior instead of relying on small-sample tables.
To find the critical value for the correlation, you use the formula \(r = \frac{\pm z}{\sqrt{n-1}}\). The \(z\) value corresponds to the critical value from the standard normal distribution table, depending on the chosen significance level \(\alpha\). Essentially, \(r\) tells us when we should consider the rank correlation coefficient significantly different from zero, implying a relationship between the variables examined.
In the exercise example, for \(n = 50\), the critical value \(r = \pm 0.28\) means any correlation greater than or equal to \(+0.28\) or less than or equal to \(-0.28\) is deemed statistically significant.

Two-Tailed Test

A two-tailed test is a type of statistical hypothesis test where deviations in two directions from the hypothesized parameter value are considered. It is commonly used when testing for non-directional hypotheses. In other words, it allows for the possibility that there could be an effect in either direction (either positive or negative).
In the given exercise, a two-tailed test is used. This means we are interested in deviations both above and below a critical threshold, which for many standard normal distribution applications aids in detecting any significant departure from a null hypothesis value, not just in a single direction.

• Lower tail: Tests for values significantly smaller than the expected parameter.
• Upper tail: Tests for values significantly larger than the expected parameter.

For a significance level \(\alpha = 0.05\), the total probability of making a Type I error (rejecting a true null hypothesis) is spread across both tails of the distribution, implying each tail has 2.5% (\(0.025\)) of the total error probability.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a vital concept in statistics often characterized by its bell-shaped curve. This distribution is symmetric around its mean, and its shape is determined by the mean and standard deviation.
For large sample sizes, sample distributions of statistics (like the rank correlation coefficient) tend to follow a normal distribution, a result of the Central Limit Theorem. In the given textbook solution, a normal distribution is used to approximate the sampling distribution of the rank correlation coefficient for the hypothesis test, which allows the test to utilize critical values derived from the standard normal distribution. This is usually acceptable when the sample size \(n \ge 30\).
The standard normal distribution is a special form of the normal distribution with a mean of 0 and a standard deviation of 1, and tables of \(z\)-scores are used as a reference for determining critical values in many statistical tests.

Significance Level

The significance level \(\alpha\) is a threshold used in hypothesis testing that signifies the probability of rejecting the null hypothesis when it is true, also known as a Type I error. It reflects how much risk you are willing to accept for this type of error.
In the given textbook exercise, the significance level is set at \(\alpha = 0.05\), which is a common choice. This means that there is a 5% risk of concluding that an effect exists when it does not.

• \(\alpha = 0.05\): Indicates that 5% of the time, you would expect to reject a true null hypothesis, or in other words, tolerate a 5% chance of making a false positive conclusion.
• Commonly accepted across many studies and fields, \(\alpha = 0.05\) strikes a balance between being too lenient and being overly strict.

The significance level impacts the critical values, as it determines how much deviation from the null hypothesis is observed to be significant. For two-tailed tests, the \(\alpha\) level must be split between the two tails, thereby affecting the critical threshold \(z\) values extracted from the normal distribution.