Agriculture: Wheat Rothamsted Experimental Station

Statistical Literacy When using the F distribution to test variances from two populations, should the random variables from each population be independent or dependent?

Statistical Literacy When using the F distribution to test two variances, is it essential that each of the two populations be normally distributed? Would it be all right if the populations had distributions that were mound-shaped and more or less symmetrical?

Statistical Literacy In general, is the F distribution symmetrical? Can values of the F distribution be negative?

Statistical Literacy To use the F distribution, what degrees of freedom need to be calculated?

Agriculture: Wheat Rothamsted Experimental Station (England) has studied wheat production since 1852. Each year, many small plots of equal size but different soil/fertilizer conditions are planted with wheat. At the end of the growing season, the yield (in pounds) of the wheat on the plot is measured. The following data are based on information taken from an article by G. A. Wiebe in the Journal of Agricultural Research (Vol. 50, pp. 331–357). For a random sample of years, one plot gave the following annual wheat production (in pounds): Use a calculator to verify that, for this plot, the sample variance is \(s^2 \approx 0.332\). Another random sample of years for a second plot gave the following annual wheat production (in pounds): Use a calculator to verify that the sample variance for this plot is \(s^2 \approx 0.089\). Test the claim that the population variance of annual wheat production for the first plot is larger than that for the second plot. Use a 1% level of significance.

Agriculture: Wheat Two plots at Rothamsted Experimental Station (see reference in Problem 5) were studied for production of wheat straw. For a random sample of years, the annual wheat straw production (in pounds) from one plot was as follows: Use a calculator to verify that, for the preceding data, \(s^2 \approx 0.318\). Another random sample of years for a second plot gave the following annual wheat straw production (in pounds): Use a calculator to verify that, for these data, \(s^2 \approx 1.078\). Test the claim that there is a difference (either way) in the population variance of wheat straw production for these two plots. Use a 5% level of significance.

Economics: Productivity An economist wonders if corporate productivity in some countries is more volatile than in other countries. One measure of a companys 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: 4.4 5.2 3.7 3.1 2.5 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 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: 3.0 3.6 3.7 4.5 5.1 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 s 2 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 economists question regarding volatility (data spread) of corporate productivity of large companies in France compared with companies in Germany? 8

Economics: Productivity A random sample of leading companies in South Korea gave the following percentage yields based on assets (see reference in Problem 7): 2.5 2.0 4.5 1.8 0.5 3.6 2.4 0.2 1.7 1.8 1.4 5.4 1.1 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: 2.3 3.2 3.6 1.2 3.6 2.8 2.3 3.5 2.8 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 economists question regarding volatility of corporate productivity of large companies in South Korea compared with those in Sweden?

Investing: Mutual Funds You dont need to be rich to buy a few shares in a mutual fund. The question is, how reliable are mutual funds as investments? This depends on the type of fund you buy. The following data are based on information taken from Morningstar, a mutual fund guide available in most libraries. A random sample of percentage annual returns for mutual funds holding stocks in aggressive-growth small companies is shown below. 1.8 14.3 41.5 17.2 16.8 4.4 32.6 7.3 16.2 2.8 34.3 10.6 8.4 7.0 2.3 18.5 25.0 9.8 7.8 24.6 22.8 Use a calculator to verify that s 2
348.43 for the sample of aggressive-growth small company funds. Another random sample of percentage annual returns for mutual funds holding value (i.e., market underpriced) stocks in large companies is shown below. 16.2 0.3 7.8 1.6 3.8 19.4 2.5 15.9 32.6 22.1 3.4 0.5 8.3 25.8 4.1 14.6 6.5 18.0 21.0 0.2 1.6 622 Use a calculator to verify that s 2
137.31 for value stocks in large companies. Test the claim that the population variance for mutual funds holding aggressive-growth small stocks is larger than the population variance for mutual funds holding value stocks in large companies. Use a 5% level of significance. How could your test conclusion relate to the question of reliability of returns for each type of mutual fund?

Investing: Mutual Funds How reliable are mutual funds that invest in bonds? Again, this depends on the bond fund you buy (see reference in Problem 9). A random sample of annual percentage returns for mutual funds holding short term U.S. government bonds is shown below. Use a calculator to verify that \(s^2 \approx 13.59\) for the preceding data. A random sample of annual percentage returns for mutual funds holding intermediate-term corporate bonds is shown below. Use a calculator to verify that \(s^2 \approx 72.06\) for returns from mutual funds holding intermediate-term corporate bonds. Use \(\alpha = 0.05\) to test the claim that the population variance for annual percentage returns of mutual funds holding short-term government bonds is different from the population variance for mutual funds holding intermediate-term corporate bonds. How could your test conclusion relate to the question of reliability of returns for each type of mutual fund?

Engineering: Fuel Injection A new fuel injection system has been engineered for pickup trucks. The new system and the old system both produce about the same average miles per gallon. However, engineers question which system (old or new) will give better consistency in fuel consumption (miles per gallon) under a variety of driving conditions. A random sample of 31 trucks were fitted with the new fuel injection system and driven under different conditions. For these trucks, the sample variance of gasoline consumption was 58.4. Another random sample of 25 trucks were fitted with the old fuel injection system and driven under a variety of different conditions. For these trucks, the sample variance of gasoline consumption was 31.6. Test the claim that there is a difference in population variance of gasoline consumption for the two injection systems. Use a 5% level of significance. How could your test conclusion relate to the question regarding the consistency of fuel consumption for the two fuel injection systems?

Engineering: Thermostats A new thermostat has been engineered for the frozen food cases in large supermarkets. Both the old and new thermostats hold temperatures at an average of \(25^{\circ}F\). However, it is hoped that the new thermostat might be more dependable in the sense that it will hold temperatures closer to \(25^{\circ}F\). One frozen food case was equipped with the new thermostat, and a random sample of 21 temperature readings gave a sample variance of 5.1. Another similar frozen food case was equipped with the old thermostat, and a random sample of 16 temperature readings gave a sample variance of 12.8. Test the claim that the population variance of the old thermostat temperature readings is larger than that for the new thermostat. Use a 5% level of significance. How could your test conclusion relate to the question regarding the dependability of the temperature readings?