Question

What price do farmers get for their watermelon crops? In the third week of July, a random sample of 42 farming regions gave a sample mean of x bar = $6.88 per 100 pounds of watermelon. Assume that σ is known to be $1.94 per 100 pounds.

(a) Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop (in dollars). What is the margin of error (in dollars)? (For each answer, enter a number. Round your answers to two decimal places.) lower limit $ upper limit $ margin of error $

(b) Find the sample size necessary for a 90% confidence level with maximal error of estimate E = 0.27 for the mean price per 100 pounds of watermelon. (Enter a number. Round up to the nearest whole number.) farming regions

(c) A farm brings 15 tons of watermelon to market. Find a 90% confidence interval for the population mean cash value of this crop (in dollars). What is the margin of error (in dollars)? Hint: 1 ton is 2000 pounds. (For each answer, enter a number. Round your answers to two decimal places.) lower limit $ upper limit $ margin of error $

Answer 1

a)

Level of Significance ,    α =    0.1          
'   '   '          
z value=   z α/2=   1.645   [Excel formula =NORMSINV(α/2) ]      
                  
Standard Error , SE = σ/√n =   1.94/√42=   0.2993          
margin of error, E=Z*SE =   1.6449   *   0.2993   =   0.49
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    6.88   -   0.4924   =   6.39
Interval Upper Limit = x̅ + E =    6.88   -   0.4924   =   7.37
90%   confidence interval is (   6.39   < µ <   7.37   )

b)

Standard Deviation ,   σ =    1.94                  
sampling error ,    E =   0.27                  
Confidence Level ,   CL=   90%                  
                          
alpha =   1-CL =   10%                  
Z value =    Zα/2 =    1.645   [excel formula =normsinv(α/2)]              
                          
Sample Size,n = (Z * σ / E )² = (   1.645   *   1.94   /   0.27   ) ² =   139.679
                          
                          
So,Sample Size needed=       140                  

c)

margin of error, E=Z*SE =      0.49*300 = 147.72
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    6.39*300 = 1916.28
Interval Upper Limit = x̅ + E =     7.37*300=2211.72