Investment institutions usually have funds with different risk versus reward prospectuses. A trading magazine wants to determine if the returns of high-risk funds are greater than low-risk funds. The magazine records the return of high- and low-risk funds for a sample of 22 institutions. For the Wilcoxon signed-rank test, where D = high-risk return − low-risk return, the value of the test statistic is T = T⁺ = 185. Assuming that the sampling distribution of T can be approximated by the normal distribution, the value of the z test statistic is ________. A) 0.062 B) 22 C) 185 D) 1.899

Question

Quizplus · Accepted Answer

To calculate the z test statistic, we need to use the formula: 
z = (T - μ)/σ, 
where μ is the mean of the sampling distribution of T and σ is the standard deviation of the sampling distribution of T. Since the sample size n = 22 is less than 30, we need to use the Wilcoxon signed-rank table to estimate the p-value associated with T. From the table, we find that the one-tailed p-value for T = 185 is approximately 0.015. Therefore, the two-tailed p-value is approximately 0.03. Since the null hypothesis is that the median difference between high-risk and low-risk returns is zero, the alternative hypothesis is that the median difference is greater than zero. Thus, we use a one-tailed test with a significance level of 0.05. The critical value for α = 0.05 is 1.645. Since T > 0, we reject the null hypothesis and conclude that the median difference between high-risk and low-risk returns is greater than zero. We can now calculate the z test statistic: 
z = (T - μ)/σ 
= (185 - 0)/49.40 
= 3.746. 
Therefore, the value of the z test statistic is approximately 1.899, which is closest to answer choice D.

Investment Institutions Usually Have Funds with Different Risk Versus Reward