Question

Two catalysts in a batch chemical process are being compared for their effect on the output of the process reaction. A sample of 11 batches was prepared using catalyst 1 and gave an average yield of 76 with a sample standard deviation of 6. A sample of 15 batches was prepared using catalyst 2 and gave an average yield of 70 and a sample standard deviation of 3. Find a 90​% confidence interval for the difference between the population​ means, assuming that the populations are approximately normally distributed with equal variances.

Answer 1

Pooled standard deviation = sqrt [ (n1-1) S21 + (n2-1) S22 / (n1+ n2 -2) ]

= sqrt [ 10 * 62 + 14 * 32 / (11 + 15 - 2) ]

= 4.5

df = n1 + n2 - 2 = 11 + 15 - 2 = 24

t critical value at 0.10 significance level with 24 df = 1.711

margin of error E = t * S / sqrt [ 1 / n1 + 1 / n2 ]

= 1.711 * 4.5 / sqrt [ 1/11 + 1/15 ]

= 3.0564

90% CI is

(1 - 2 ) - E < 1 - 2 < (1 - 2 ) + E

(76 - 70) - 3.0564 < 1 - 2 < (76 - 70) + 3.0564

2.9436 < 1 - 2 < 9.0564

90% CI is ( 2.9436 , 9.0564 )