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An economist wonders if corporate productivity in some countries is more volatile than in other countries. One measure of a company's productivity is annual percentage yield based on total company assets. Data for this problem are based on information taken from Forbes Top Companies, edited by J. T. Davis. A random sample of leading companies in France gave the following percentage yields based on assets: \(\begin{array}{llllllllllll}4.4 & 5.2 & 3.7 & 3.1 & 2.5 & 3.5 & 2.8 & 4.4 & 5.7 & 3.4 & 4.1\end{array}\) \(\begin{array}{llllllllll}6.8 & 2.9 & 3.2 & 7.2 & 6.5 & 5.0 & 3.3 & 2.8 & 2.5 & 4.5\end{array}\) Use a calculator to verify that \(s^{2}=2.044\) for this sample of French companies. Another random sample of leading companies in Germany gave the following percentage yields based on assets: \(\begin{array}{rrrrrrrrr}3.0 & 3.6 & 3.7 & 4.5 & 5.1 & 5.5 & 5.0 & 5.4 & 3.2\end{array}\) \(\begin{array}{llllllllll}3.5 & 3.7 & 2.6 & 2.8 & 3.0 & 3.0 & 2.2 & 4.7 & 3.2\end{array}\) Use a calculator to verify that \(s^{2} \approx 1.038\) for this sample of German companies. Test the claim that there is a difference (either way) in the population variance of percentage yields for leading companies in France and Germany. Use a \(5 \%\) level of significance. How could your test conclusion relate to the economist's question regarding volatility (data spread) of corporate productivity of large companies in France compared with companies in Germany?

Step by step solution

01

State the hypotheses

We need to test if there is a significant difference in the variances of the two samples (France and Germany). The null hypothesis \( H_0 \) is that the variances are equal \( \sigma_1^2 = \sigma_2^2 \) and the alternative hypothesis \( H_a \) is that the variances are not equal \( \sigma_1^2 eq \sigma_2^2 \).

02

Calculate the F-statistic

To compare the variances, use the F-statistic: \( F = \frac{s_1^2}{s_2^2} \) where \( s_1^2 = 2.044 \) is the variance for France and \( s_2^2 = 1.038 \) is the variance for Germany. So, \( F = \frac{2.044}{1.038} \approx 1.970 \).

03

Determine degrees of freedom

The degrees of freedom \( df_1 \) for the French sample is \( n_1 - 1 = 21 - 1 = 20 \) and for the German sample \( df_2 = n_2 - 1 = 18 - 1 = 17 \). These will be used to find the critical value from the F-distribution table.

04

Find the critical value

Using an F-distribution table and a significance level \( \alpha = 0.05 \), we find the critical values for \( F_{20,17} \). Checking a table or calculator, the critical value for a two-tailed test at \( \alpha/2 = 0.025 \) is approximately between 0.405 and 2.54.

05

Compare F-statistic to critical value

Our calculated F-statistic is \( 1.970 \). Since \( 1.970 \) is less than the upper critical value \( 2.54 \) and greater than the lower critical value \( 0.405 \), we do not reject the null hypothesis.

06

Make the conclusion

Since we failed to reject the null hypothesis, there is not enough evidence to claim that there is a significant difference in the variances of yields between companies in France and Germany. Thus, there is no significant evidence to suggest different levels of volatility in productivity between the two countries.

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Key Concepts

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

F-distribution

When comparing variances from two different populations, such as French and German companies, the F-distribution plays a crucial role. This distribution is widely used in hypothesis testing to determine if there are significant differences between variances of two populations. It is named after Ronald Fisher.

• The F-distribution is skewed to the right and defined only for positive values. This makes it suitable for ratios of variances since variances are always non-negative.
• The shape of the F-distribution depends on two sets of degrees of freedom: one for the numerator (the first sample variance) and another for the denominator (the second sample variance).
• When utilizing an F-distribution, you compare the calculated F-statistic to a critical value obtained from an F-distribution table, considering the desired level of significance.

This distribution helps determine whether the observed ratio of variances could occur by random chance, making it key in variance analysis.

Variance

Variance is a statistical measurement that describes the spread of data points in a dataset. Specifically, it quantifies how much the data spreads around the mean.

• A higher variance indicates that the data points are more spread out from the mean, showing more diversity or variability.
• A lower variance suggests that the data points are closer to the mean, indicating less variability.

In the context of the exercise, comparing the variances of French and German companies helps understand the consistency of their annual percentage yields. Calculating variance involves squaring the standard deviation of a dataset. For example, the variance for French companies is calculated as 2.044, and for German companies, it is approximately 1.038. This shows how variation occurs in terms of productivity within these economies, which can reflect on how stable or volatile they are.

Level of Significance

The level of significance, indicated as \( \alpha \), is a threshold in hypothesis testing that determines when you will reject the null hypothesis. It represents the probability of committing a Type I error, which occurs when you wrongly reject a true null hypothesis.

• Commonly, levels of significance are set at \( 0.05 \) or \( 0.01 \), indicating a 5% or 1% risk, respectively, of being wrong when rejecting the null hypothesis.
• The alpha level divides the F-distribution, helping isolate the critical region where we would decide to reject the null hypothesis.
• In our case, an \( \alpha \) level of 0.05 was used, implying a 95% confidence that any observed effect is not due to random chance.

Choosing the right level of significance is crucial because it balances the risk of false rejection of the null hypothesis against the risk of failing to detect a true difference.