An agricultural field trial compares the yield of two varieties of tomato for co

Question

An agricultural field trial compares the yield of two varieties of tomato for commercial use. The researchers divide in half each of 10 small plots of land in different locations and plant each tomato variety on half of each plot. After harvest they compare the yields in pounds per plant at each location. The 10 differences (Variety. Variety B) gives Z = 0.34. Assume a population standard deviation of = 0.83. We want to determine if there is a difference in the mean yields, so our null hypothesis is that there is no difference in the mean yields, ie, Ho: = 0; Ha: 0. We asked, "Is there convincing evidence at the 95% level that the mean yield of tomatoes is different from = 0?" A significance test to answer this question was carried out. The test statistics is z 1.2954 and the p-value is 0.1952. a.) Describe what a Type I error would be in this situation b.) Calculate the probability of a Type I error for this problem. c.) Describe what a Type II error would be in this situation. d.) The researchers who carried out the field trial suspect that the large p-value is due to low power. They would like to be able to detect a mean difference in yield of 0.5 pound per plant at = .05 significance level. Find the probability of a Type II error when in fact ,- 5. Then find the power of the test against the alternative = 0.5.

Explanation / Answer

a) Here Type I error would happen, if we reject the null hypothesis though it is true.

b) probability of type I error is 0.05 (this is always the significance level of the test)

c) Type II would happen, if we fail to reject the null hypothesis though in real we should have rejected it.
In the context of the problem, though the real difference in mean is not zero we fail to reject the null hypothesis.

d)

mu0 (hypothesised mean) 0 sigma 0.83 n 10 alpha 0.05 sample/true mean 0.5 Std. Error. SE = sigma/sqrt(n) 0.2625 Zcritical -1.960 Xcritical values -0.51 0.51 Beta or type II error is the probability of fail to reject the null hypothesis P(X<-0.51|mu=0.5) 0.00006 P(X>0.51|mu=0.5) 0.47808 Hence type II error probability is 0.4781 Power of the test 0.5219

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