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.A random sample of leading companies in France gave the following percentage yields based on assets. 4.6 5.4 3.4 3.1 2.4 3.5 2.8 4.4 5.7 3.4 4.1 6.8 2.9 3.2 7.2 6.5 5.0 3.3 2.8 2.5 4.5 Use a calculator to verify that the sample variance is s2 ≈ 2.109 for this sample of French companies.Another random sample of leading companies in Germany gave the following percentage yields based on assets. 3.3 3.2 3.2 4.1 5.4 5.5 5.0 5.4 3.2 3.5 3.7 2.6 2.8 3.0 3.0 2.2 4.7 3.2 Use a calculator to verify that s2 ≈ 1.077 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? (a) What is the level of significance? State the null and alternate hypotheses.a) Ho: σ12 = σ22; H1: σ12 > σ22b) Ho: σ12 > σ22; H1: σ12 = σ22 c) Ho: σ22 = σ12; H1: σ22 > σ12d) Ho: σ12 = σ22; H1: σ12 ≠ σ22 (c) Find or estimate the P-value of the sample test statistic. (Use 4 decimal places.)a) p-value > 0.200b) 0.100 < p-value < 0.200 c) 0.050 < p-value < 0.100d) 0.020 < p-value < 0.050e) 0.002 < p-value < 0.020f) p-value < 0.002 (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?a) At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.b) At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. c) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.d) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. (e) Interpret your conclusion in the context of the application.a) Fail to reject the null hypothesis, there is sufficient evidence that the variance in percentage yields on assets is greater in the French companies.b) Reject the null hypothesis, there is insufficient evidence that the variance in percentage yields on assets is greater in the French companies. c) Reject the null hypothesis, there is sufficient evidence that the variance in percentage yields on assets is different in both companies.d) Fail to reject the null hypothesis, there is insufficient evidence that the variance in percentage yields on assets is different in both companies.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
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.
A random sample of leading companies in France gave the following percentage yields based on assets.
| 4.6 | 5.4 | 3.4 | 3.1 | 2.4 | 3.5 | 2.8 | 4.4 | 5.7 | 3.4 | 4.1 |
| 6.8 | 2.9 | 3.2 | 7.2 | 6.5 | 5.0 | 3.3 | 2.8 | 2.5 | 4.5 |
Use a calculator to verify that the sample variance is s2 ≈ 2.109 for this sample of French companies.
Another random sample of leading companies in Germany gave the following percentage yields based on assets.
| 3.3 | 3.2 | 3.2 | 4.1 | 5.4 | 5.5 | 5.0 | 5.4 | 3.2 |
| 3.5 | 3.7 | 2.6 | 2.8 | 3.0 | 3.0 | 2.2 | 4.7 | 3.2 |
Use a calculator to verify that s2 ≈ 1.077 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?
a) Ho: σ12 = σ22; H1: σ12 > σ22
b) Ho: σ12 > σ22; H1: σ12 = σ22
c) Ho: σ22 = σ12; H1: σ22 > σ12
d) Ho: σ12 = σ22; H1: σ12 ≠ σ22
a) p-value > 0.200
b) 0.100 < p-value < 0.200
c) 0.050 < p-value < 0.100
d) 0.020 < p-value < 0.050
e) 0.002 < p-value < 0.020
f) p-value < 0.002
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?
a) At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
b) At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
c) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
d) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
(e) Interpret your conclusion in the context of the application.
a) Fail to reject the null hypothesis, there is sufficient evidence that the variance in percentage yields on assets is greater in the French companies.
b) Reject the null hypothesis, there is insufficient evidence that the variance in percentage yields on assets is greater in the French companies.
c) Reject the null hypothesis, there is sufficient evidence that the variance in percentage yields on assets is different in both companies.
d) Fail to reject the null hypothesis, there is insufficient evidence that the variance in percentage yields on assets is different in both companies.
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