91Ó°ÊÓ

A random sample of leading companies in South Korea gave the following percentage yields based on assets (see reference in Problem 7): \(\begin{array}{lllllll}2.5 & 2.0 & 4.5 & 1.8 & 0.5 & 3.6 & 2.4\end{array}\) \(\begin{array}{llllll}0.2 & 1.7 & 1.8 & 1.4 & 5.4 & 1.1\end{array}\) Use a calculator to verify that \(s^{2}=2.247\) for these South Korean companies. Another random sample of leading companies in Sweden gave the following percentage yields based on assets: \(\begin{array}{lllllllll}2.3 & 3.2 & 3.6 & 1.2 & 3.6 & 2.8 & 2.3 & 3.5 & 2.8\end{array}\) Use a calculator to verify that \(s^{2}=0.624\) for these Swedish companies. Test the claim that the population variance of percentage yields on assets for South Korean companies is higher than that for companies in Sweden. Use a \(5 \%\) level of significance. How could your test conclusion relate to an economist's question regarding volatility of corporate productivity of large companies in South Korea compared with that in Sweden?

Step by step solution

01

Define the Hypotheses

We are tasked with testing if the variance of percentage yields for South Korean companies is higher than that for Swedish companies. Thus, we set up our null hypothesis as \( H_0: \sigma^2_1 \leq \sigma^2_2 \) and the alternative hypothesis as \( H_a: \sigma^2_1 > \sigma^2_2 \), where \( \sigma_1^2 \) and \( \sigma_2^2 \) are the population variances for South Korea and Sweden, respectively.

02

Identify the Test Statistic

We will use the F-test for comparing two variances. The test statistic is given by \( F = \frac{s^2_1}{s^2_2} \), where \( s^2_1 = 2.247 \) and \( s^2_2 = 0.624 \) are the sample variances for South Korea and Sweden, respectively.

03

Calculate the Test Statistic

Substitute the values of the variances into the formula: \[ F = \frac{2.247}{0.624} \approx 3.601 \].

04

Determine the Critical Value

The degrees of freedom for South Korea's sample is \( n_1 - 1 = 12 - 1 = 11 \) and for Sweden's sample is \( n_2 - 1 = 9 - 1 = 8 \). Using an F-distribution table and a 5% level of significance for a one-tailed test with these degrees of freedom, the critical value \( F_{\text{critical}} \approx 3.326 \).

05

Compare Test Statistic to Critical Value

Our calculated \( F \approx 3.601 \) exceeds the critical value \( F_{\text{critical}} \approx 3.326 \).

06

Make a Decision

Since the test statistic exceeds the critical value, we reject the null hypothesis. There is sufficient evidence to claim that the variance in percentage yields based on assets for South Korean companies is higher than that for Swedish companies.

07

Relate to Economist's Question

This result suggests higher volatility in corporate productivity among large companies in South Korea compared to Sweden, as indicated by greater variability in percentage yields on assets.

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Key Concepts

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

F-test

The F-test is a statistical test used when you want to compare two variances.
It is particularly useful in determining if one population has a significantly larger variance than another.
This test involves calculating a test statistic by dividing the variance of the first sample by the variance of the second.
For example, in assessing the variance of percentage yields between South Korean and Swedish companies, the F-test statistic is calculated as \( F = \frac{s^2_1}{s^2_2} \), where \( s^2_1 \) and \( s^2_2 \) are the sample variances.
This ratio tells us how much larger one variance is compared to another. Basically, the F-test lets you check if the observed variation is larger than what could be expected by random chance.
One should select the appropriate formula and test, depending upon the structure of the problem.

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics that is used to make inferences about populations based on sample data.
In simple terms, it helps to determine if there is enough evidence in a sample to infer that a certain condition is true for the entire population.
Initially, we set up two hypotheses: the null hypothesis and the alternative hypothesis.

• The null hypothesis (\(H_0\)) usually states that there is no effect or no difference, and it assumes the status quo.
• The alternative hypothesis (\(H_a\)) is what we want to prove, claiming a true effect or difference.

The hypothesis test involves calculating a test statistic, comparing it to a critical value, and making a decision.
If the test statistic is beyond the critical value, we reject the null hypothesis.
In the case of comparing variances, as in our example, we test if the variance of the South Korean companies' percentage yields is greater than that of the Swedish companies.

Population Variance

The concept of population variance is crucial in statistics as it measures the extent to which the values of a data set differ from the mean.
When we talk about variance, we're talking about the average of the squared differences from the mean, essentially quantifying the data's spread.
In this context, variance can indicate how much percentage yields fluctuate for different countries' companies.

• A higher variance means the data points are more spread out from the mean, indicating higher volatility.
• A lower variance indicates that the data points are closer to the mean, suggesting stability.

When conducting a comparative analysis like in the South Korean versus Swedish company exercise, understanding population variance helps evaluate which country exhibits more financial stability.

Critical Value

The critical value in hypothesis testing is a threshold that the test statistic must exceed to reject the null hypothesis.
This value helps in deciding whether to accept or reject our initial hypothesis.
It's derived based on the chosen level of significance, often 5%, and the degrees of freedom associated with the data.
In an F-test, the critical value is obtained from an F-distribution table and determines the boundary of acceptance.
For our example, comparing variances of South Korean and Swedish companies, the critical value is obtained for specific degrees of freedom related to the sample sizes, specifically 11 and 8.
If the test statistic exceeds this critical value, as it did at an \( F \approx 3.601 \), it indicates sufficient evidence to reject the null hypothesis and support that one variance is larger than the other.