#### 20. Comparing Two Means

*Question 1*

The uncoated leaves have a mean uptake of 143.8 mm with standard deviation 11.024 mm while the coated leaves have a mean uptake of 103.6 mm with standard deviation of 10.645 mm. This gives an estimated difference of 40.2 mm. Since the standard deviations are similar it is reasonable to pool them together. The pooled variance is

(19×11.024^{2} + 19×10.645^{2})/38 = 117.422,

giving a pooled standard deviation of 10.836 mm. This estimate has 38 degrees of freedom. Table 17.2 doesn't have a row for 38 so we use a conservative critical value for 95% confidence from the df=30 row, *t*^{*} = 2.042. This gives a margin of error of

2.042 × 10.836 √(1/20 + 1/20) = 6.997.

Thus the confidence interval is given by 40.2 ± 7.0 mm, or (33.2, 47.2) mm.