2 A farmer is worried about the sweetness of his oranges fro

2. A farmer is worried about the sweetness of his oranges from a particular grove. While oranges can never be too sweet by his account, low sugar content is had for business. He randomly samples 30 oranges from the grove and determines that their average sugar contents is 0.025 lb., the standard deviation of the 30 oranges is 0.003 lb. Should he be worried, if his other groves produce oranges with sugar content of 0.027 lb. per orange? Assume that the sugar content of the oranges is normally distributed.

Solution

As the level of significance is not given, we assume it is 0.05.

Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   0.027  
Ha:    u   <   0.027  
              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical z, as alpha =    0.05   ,      
alpha =    0.05          
zcrit =    -   1.644853627      
              
Getting the test statistic, as              
              
X = sample mean =    0.025          
uo = hypothesized mean =    0.027          
n = sample size =    30          
s = standard deviation =    0.003          
              
Thus, z = (X - uo) * sqrt(n) / s =    -3.651483717          
              
Also, the p value is              
              
p =    0.000130365          
              
Comparing |z| > 1.645 (or, p < 0.05), we   REJECT THE NULL HYPOTHESIS.          
              
Thus, there is significant evidence that the sugar content of the said grove is less than the sugar content of the other groves, at 0.027 lb per orange. [CONCLUSION]