Statistics for Environmental Engineers

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Because these are greater than 5, the normal approximation can be used. The sample size is large enough that the Yates continuity correction is not needed.

Might the true difference of the two proportions be zero? Could we observe a difference as large as 0.1 just as a result of random variation? This can be checked by examining the lower 95% confidence limit. From the observed proportions, we estimate p = (0.9 + 0.8)/2 = 0.85 and 1 — p = 0.15. For a onesided test at the 95% level, za=o.o5 = 1.645 and the lower 95% confidence bound is:

(0.9-0.8) — 1.645 J2(a15e°’85 = 0.1 — 1.645(0.056) = 0.007

The lower limit is greater than zero, so it is concluded that the observed difference is larger than is expected to occur by chance and that p2 = 0.8 is less than p1 = 0.9.

Alternately, we could have compared a computed sample z-statistic:

1.77

z =

0.90-0.80 2 (0.15)(0.85) 80

with the tabulated value of z0.05 = 1.645. Finding zsample = 1.77 > z0.05 = 1.645 means that the observed difference is quite far onto the tail of the distribution. It is concluded that the difference between the proportions is large enough to say that the two treatments are different.

Comments

The example problem found a small difference in proportions to be statistically significant because the number of test organisms was large. Many bioassays use about 20 test organisms, in which case the observed difference in proportions will have to be quite large before we can have a high degree of confidence that a difference did not arise merely from chance.

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