Question

The average yield of the standard variety off soybeans per acre on a farm in a region is 48.8 bushels per acre, with a standard deviation of 12.0 bushels per acre. We have developed a new variety of soybean that we believe will have a higher mean yield per acre. To test this, we draw a random sample of 36 one-acre plots from the region, plant the new variety of soybean, and compute the sample mean yield per acre. The sample mean yield per acre is 53.4 bushels. Carry out the test at the 0.01 level of significance. (You can assume that yields per acre are approximately normally distributed.) Follow all steps clearly and write a clear conclusion. Include all necessary conditions check and show all your working.

Answer 1

Let X be the average yield of soyabean      
μ = 48.8 Population Mean     
σ = 12 Population Standard Deviation     
X̅ = 53.4 Sample Mean     
n = 36 Sample Size     
Let α = 0.01 Level of significance     
Since population standard deviation is known, we use the z-test      
The null and alternative hypothesis are      
Ho : μ = 48.8      
H1 : μ > 48.8      
       
Find Calculated Value of test statistic z      

z-score = 2.3      
This is a right sided z-test      
p-value for the test is found using Excel function NORM.S.DIST      
p-value = 1-NORM.S.DIST(2.3, TRUE)         (For right sided test, subtract from 1)
p-value = 0.0107      
       
Interpretation of p-value      
Since 0.0107 > 0.01                                   (p-value is marginally higher than significance level)    
that is p-value > α      
we DO NOT reject Ho      
       
Conclusion :      
There does not exist sufficient statistical evidence to conclude that       
new variety of soybean will have a higher mean yield per acre      
       

(x- μ) (53.4-48.8) 12 oWn V36