Applied Statistics and Probability for Engineers

346 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

EXAMPLE 10-8 An article in the journal Hazardous Waste and Hazardous Materials(Vol. 6, 1989) reported

the results of an analysis of the weight of calcium in standard cement and cement doped

with lead. Reduced levels of calcium would indicate that the hydration mechanism in the

cement is blocked and would allow water to attack various locations in the cement struc-

ture. Ten samples of standard cement had an average weight percent calcium of

with a sample standard deviation of s 1 5.0, while 15 samples of the lead-doped cement

had an average weight percent calcium of with a sample standard deviation of

s 2 4.0.

We will assume that weight percent calcium is normally distributed and find a 95% con-

fidence interval on the difference in means,  1  2 , for the two types of cement. Furthermore,

we will assume that both normal populations have the same standard deviation.

The pooled estimate of the common standard deviation is found using Equation 10-12 as

follows:

Therefore, the pooled standard deviation estimate is The 95% confi-

dence interval is found using Equation 10-19:

or upon substituting the sample values and using t0.025,232.069,

which reduces to

Notice that the 95% confidence interval includes zero; therefore, at this level of confidence we

cannot conclude that there is a difference in the means. Put another way, there is no evidence

that doping the cement with lead affected the mean weight percent of calcium; therefore, we

cannot claim that the presence of lead affects this aspect of the hydration mechanism at the

95% level of confidence.

Case 2: ^21 ^22

In many situations it is not reasonable to assume that ^21 ^22. When this assumption is un-

warranted, we may still find a 100(1)% confidence interval on  1  2 using the fact

that is distributed approximately as twith

degrees of freedom vgiven by Equation 10-16. The CI expression follows.

T* 3 X 1 X 2  1  1  224
̨ 2 S^21 n 1 S^22 n 2

0.72 1  2 6.72

90.087.0 2.069 1442

B

1

10

1

15

90.087.02.069 1 4.4 2

B

1

10

1

15

 1  2

x 1 x 2 t0.025,23 sp

B

1

n 1

1

n 2 ^1 ^2 x^1 x^2 t0.025,23 sp^ B

1

n 1

1

n 2

sp 1 19.524.4.

19.52



91 5.0 22141 4.0 22

1015  2

s^2 p

1 n 1  12 s^211 n 2  12 s^22

n 1 n 2  2

x 2 87.0,

x 1 90.0,

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