91Ó°ÊÓ

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Compute the sample test statistic. (c) Find or estimate the \(P\) -value of the sample test statistic. (d) Conclude the test. (e) Interpret the conclusion in the context of the application. Economics: Stocks As an economics class project, Debbie studied a random sample of 14 stocks. For each of these stocks, she found the cost per share (in dollars) and ranked each of the stocks according to cost. After 3 months, she found the earnings per share for each stock (in dollars). Again, Debbie ranked each of the stocks according to earnings. The way Debbie ranked, higher ranks mean higher cost and higher earnings. The results follow, where \(x\) is the rank in cost and \(y\) is the rank in earnings. $$\begin{array}{|l|cccccccccccccc} \hline \text { Stock } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ \hline x \text { rank } & 5 & 2 & 4 & 7 & 11 & 8 & 12 & 3 & 13 & 14 & 10 & 1 & 9 & 6 \\ \hline y \text { rank } & 5 & 13 & 1 & 10 & 7 & 3 & 14 & 6 & 4 & 12 & 8 & 2 & 11 & 9 \\ \hline \end{array}$$ Using a 0.01 level of significance, test the claim that there is a monotone relation, cither way, between the ranks of cost and earnings.

Step by step solution

01

Determine Level of Significance and State Hypotheses

The level of significance is given as 0.01. We need to state the null and alternate hypotheses for testing a rank correlation. **Null Hypothesis ( H_0 ):** There is no monotone relation between the cost rank and earnings rank. **Alternate Hypothesis ( H_1 ):** There is a monotone relation between the cost rank and earnings rank.

02

Calculate Differences and Squared Differences

For each pair of ranks, calculate the difference ( D_i = x_i - y_i ), where D_i is the difference, x_i is the rank of cost, and y_i is the rank of earnings. Then, compute the squared differences ( D_i^2 ). Example calculations: 1. For Stock 1: D = 5 - 5 = 0 , D^2 = 0 . This is done for all stocks.

03

Compute Test Statistic (Spearman's Rank Correlation Coefficient)

The test statistic is given by Spearman's rank correlation coefficient ( ho) formula:\[\rho = 1 - \frac{6 \sum D_i^2}{n(n^2 - 1)}\]where \(\sum D_i^2\) is the sum of the squared differences computed in Step 2, and n = 14 is the number of stocks. Insert the calculated values to compute \rho.

04

Estimate the P-value

Once we have ho , we can use statistical tables or software to estimate the P -value corresponding to the test statistic for n - 2 degrees of freedom. Compare the P -value with the level of significance (0.01).

05

Conclude the Test

If the P -value is less than the significance level (0.01), reject the null hypothesis. Otherwise, do not reject the null hypothesis. This determination will indicate whether there is a significant monotone relation between cost and earnings.

06

Interpret the Conclusion

Interpret the results in terms of the economics context. If you rejected the null hypothesis, it means there is significant evidence to suggest a monotone relationship exists between cost and earnings ranks among the stocks. If not, there is no significant evidence supporting such a relation.

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Key Concepts

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

Level of Significance

In hypothesis testing, the level of significance is a critical threshold that determines the likelihood of rejecting the null hypothesis when it is actually true. It is denoted by the Greek letter \( \alpha \).
For instance, if the level of significance is set at 0.01, there is a 1% risk of concluding that a difference exists when there is no actual difference. This threshold is chosen by researchers before conducting the test. It helps in assessing how strict the criteria are for concluding a significant result.
Key points about the level of significance:

• It is a probability value that indicates the risk of a Type I error, which is rejecting a true null hypothesis.
• Commonly used significance levels are 0.05, 0.01, and 0.10.
• A lower \( \alpha \) means stricter criteria for significance, reducing the risk of false positives.

For Debbie's stock analysis, a 0.01 significance level indicates a stringent test where only results with strong evidence against the null hypothesis will be considered significant.

Spearman's Rank Correlation Coefficient

Spearman's Rank Correlation Coefficient helps in measuring the strength and direction of association between two ranked variables. It is a non-parametric test, meaning it's distribution-free, and useful when the data doesn't conform to a normal distribution.
This coefficient is particularly beneficial when dealing with rank data or ordinal data, where the intervals between values are not of equal size. It works by converting raw scores to ranks and examining the disparities between them.The calculation involves:

• Rank each set of data. In Debbie's project, she ranks stocks first by cost and then by earnings.
• Compute the difference between the ranks for each item.
• Square these differences and find their sum.
• Use the formula: \[\rho = 1 - \frac{6 \sum D_i^2}{n(n^2 - 1)}\] where \( \rho \) is the Spearman's rank correlation coefficient, \( D_i \) is the difference between ranks for each pair, and \( n \) is the number of pairs.

A \( \rho \) value close to +1 or -1 indicates a strong monotone relationship, while a \( \rho \) close to 0 suggests a weak or no monotone relationship.

Null and Alternate Hypotheses

In any statistical test, defining the null and alternate hypotheses is essential. These hypotheses are mutually exclusive statements about a population:

• The **Null Hypothesis (\( H_0\))** asserts that there is no effect or no difference. In Debbie's context, it suggests that there is no monotone relation between the cost ranks and earnings ranks of the stocks.
• The **Alternate Hypothesis (\( H_1\))** posits that there is an effect or a difference. For Debbie's study, this implies that a monotone relationship does exist between the costs and earnings ranks.

The purpose of the hypothesis test is to determine which of these hypotheses is supported by the sample data.
If the test results lead to a statistically significant finding (meaning the p-value is less than the level of significance), the null hypothesis is rejected in favor of the alternative. Conversely, if the test lacks significance, the null hypothesis is not rejected.
This logical structure forms the backbone of statistical hypothesis testing, helping determine the direction of the research findings.