Question

In the “Probable Error of a Mean” by William Gossett the following example was used. Yields of corn in pounds per acre were calculated for kiln-dried seed and seed that was not dried. (It was believed in the day that kiln-dried seed produces a higher yield). The yield values for dried seed were:

2009, 1915, 2011, 2463, 2180, 1925, 2122, 1482, 1542, 1443, 1535

The yields achieved with regular seed were:

1903, 1935, 1910, 2496, 2108, 1961, 2060, 1444, 1612, 1316, 1511

Construct a 95% confidence interval estimate of the mean yield in these two cases. Do they look similar?

Answer 1

Let X represent Kiln dried seeds      
Let Y represent undried seeds      
n = sample size = 11      
Using Excel functions Average and stdev.s we find the mean and standard deviations of X and Y      
Mean X = Mx = 1875.1818      
Standard Deviation X = Sx = 332.8501      
      
Mean Y = My = 1841.4545      
Standard Deviation Y = Sy = 342.7373      
      
Confidence interval for mean is given by      

Confidence Interval for Kiln dried seeds      
For 95% confidence interval, α = 0.05, α/2 = 0.025      
From the z-tables, or Excel function NORM.S.INV(α/2)      
z = NORM.S.INV(0.025) = 1.96    (We take the positive value for calculations)  
Confidence interval is      

= (1678.48, 2071.88)      
      
95% Confidence Interval for mean yield of Kiln dried seeds = (1678.48, 2071.88)      
      
      
Confidence Interval for non dried seeds      
For 95% confidence interval, α = 0.05, α/2 = 0.025      
From the z-tables, or Excel function NORM.S.INV(α/2)      
z = NORM.S.INV(0.025) = 1.96    (We take the positive value for calculations)  
Confidence interval is      

= (1638.91, 2043.9993)      
      
95% Confidence Interval for mean yield of non dried seeds is (1638.91, 2043.9993)      
      
Although there is not a very big difference, it can be seen that since the lower limit of yield of kiln dried seeds       
is higher than the lower limit of the yields of non-dried seeds      
the kiln dried seeds produced a little higher yield than the undried seeds.