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Significance Testing Format

The most important material reclaimed from beverage bottles is PET plastic. A serious impurity is aluminum, which later can clog the filters in extruders when the recycled material is used. The following are the amounts (in ppm by weight of aluminum) found in bihourly samples of PET recovered at the plant over roughly a two day period.

291, 222, 125, 79, 145, 119, 244, 118, 182, 63, 30, 140, 101, 102, 87, 183, 60, 191, 119, 511, 120, 172, 70, 30, 90, 115

It is desirable to have mean aluminum content for samples of recycled plastic 200 ppm. Use the step-wise significance testing format to find whether this contamination goal is violated.

(a) Under variance (Var X)=1
(b) Under large sample assumption and variance (Var X) unknown
(c) Under small sample assumption and variance (Var X)unknown
(d) Give 95% confidence interval for the mean under large sample assumption
(e) Give 95% confidence interval for the mean under small sample assumption

Solution Preview

See the attached file.
(a) Under variance (Var X)=1

(Please refer to attached EXCEL for calculation)
Calculate the mean of the samples is Xm = 142.65, with sample size = 25
The standard error is SE = SQRT(Var X/ n) = SQRT(1/ 25) = 0.2
Compute z = (M - Xm)/SE = (200-142.65)/ 0.2 = 286.73
Since the z value is extremely high, (Prob(z > 286.73) = 0), we can say that the mean aluminum content for samples is significantly less than 200 ppm.

(b) Under large sample assumption and variance (Var X) unknown

Under large sample assumption the distribution is asymptotically standard normal and we can use z test.
Now ...

Solution Summary

The solution assists with using the step-wise significance testing format to find whether the contamination goal is violated.

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