EXAMPLE 7.12 Wheat Prices

The U.S. Department of Agriculture (USDA) uses sample surveys to produce important economic estimates.²² One pilot study estimated wheat prices in July and in January using independent samples of wheat producers in the two months. Here are the summary statistics, in dollars per bushel:

The July prices are higher on the average. But we have data from only a limited number of producers each month. Can we conclude that national average prices in July and January are not the same? Or are these differences merely what we would expect to see due to random variation?

Because we did not specify a direction for the difference before looking at the data, we choose a two-sided alternative. The hypotheses are

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Because the samples are moderately large, we can confidently use the procedures even though we lack the detailed data and so cannot verify the Normality condition.

The two-sample statistic is

The conservative approach finds the -value by comparing 5.16 to critical values for the distribution because the smaller sample has 45 observations. We must double the table tail area because the alternative is two-sided.

Table D does not have entries for 44 degrees of freedom. When this happens, we use the next smaller degrees of freedom. Our calculated value of is larger than the entry in the table. Doubling 0.0005, we conclude that the -value is less than 0.001. The data give conclusive evidence that the mean wheat prices were higher in July than they were January .