370 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES(a) State any assumptions necessary to test the claim that both
tips produce the same Rockwell C-scale hardness readings.
Check those assumptions for which you have the information.
(b) Apply an appropriate statistical method to determine if the
data support the claim that the difference in Rockwell
C-scale hardness readings of the two tips is significantly
different from zero
(c) Suppose that if the two tips differ in mean hardness read-
ings by as much as 1.0, we want the power of the test to be
at least 0.9. For an 0.01, how many coupons should
have been used in the test?
10-88. Two different gauges can be used to measure the depth
of bath material in a Hall cell used in smelting aluminum. Each
gauge is used once in 15 cells by the same operator.(a) State any assumptions necessary to test the claim that
both gauges produce the same mean bath depth read-
ings. Check those assumptions for which you have the
information.
(b) Apply an appropriate statistical procedure to determine if
the data support the claim that the two gauges produce dif-
ferent mean bath depth readings.
(c) Suppose that if the two gauges differ in mean bath depth
readings by as much as 1.65 inch, we want the power of
the test to be at least 0.8. For 0.01, how many cells
should have been used?
10-89. An article in the Journal of the Environmental
Engineering Division(“Distribution of Toxic Substances in
Rivers,” 1982, Vol. 108, pp. 639–649) investigates the con-
centration of several hydrophobic organic substances in the
Wolf River in Tennessee. Measurements on hexachloroben-
zene (HCB) in nanograms per liter were taken at different
depth downstream of an abandoned dump site. Data for two
depths follow:
Surface: 3.74, 4.61, 4.00, 4.67, 4.87, 5.12, 4.52, 5.29, 5.74, 5.48
Bottom: 5.44, 6.88, 5.37, 5.44, 5.03, 6.48, 3.89, 5.85, 6.85, 7.16
(a) What assumptions are required to test the claim that
mean HCB concentration is the same at both depths?
Check those assumptions for which you have the infor-
mation.
(b) Apply an appropriate procedure to determine if the data
support the claim in part a.
(c) Suppose that the true difference in mean concentrations is
2.0 nanograms per liter. For 0.05, what is the power
of a statistical test for H 0 : 1 2 versus H 1 : 1 2?
(d) What sample size would be required to detect a difference
of 1.0 nanograms per liter at 0.05 if the power must
be at least 0.9?Coupon Tip 1 Tip 2 Coupon Tip 1 Tip 2
1 47 46 6 41 41
2 42 40 7 45 46
3 43 45 8 45 46
4 40 41 9 49 48
54243Cell Gauge 1 Gauge 2 Cell Gauge 1 Gauge 2
1 46 in. 47 in. 9 52 51
250 53 10 47 45
347 45 11 49 51
453 50 12 45 45
549 51 13 47 49
648 48 14 46 43
753 54 15 50 51
856 53MIND-EXPANDING EXERCISES10-90. Three different pesticides can be used to control
infestation of grapes. It is suspected that pesticide 3 is
more effective than the other two. In a particular vineyard,
three different plantings of pinot noir grapes are selected
for study. The following results on yield are obtained:ni
(Bushels/ (Number of
Pesticide Plant) si Plants)
1 4.6 0.7 100
2 5.2 0.6 120
3 6.1 0.8 130If iis the true mean yield after treatment with the ith
pesticide, we are interested in the quantitywhich measures the difference in mean yields between
pesticides 1 and 2 and pesticide 3. If the sample sizes ni
are large, the estimator (say, ) obtained by replacing
each individual iby is approximately normal.
(a) Find an approximate 100(1 )% large-sample
confidence interval for .Xiˆ
1
2
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