Question

The rare earth element gadolinium is often used as a contrasting agent for MRIs. The concentration of gadolinium was measured in the discharge from a wastewater treatment plant near a large medical facility. Eight measurements were obtained: 1.212001.21200 ppm, 1.212001.21200 ppm, 1.209001.20900 ppm, 1.203001.20300 ppm, 1.777001.77700 ppm, 1.201001.20100 ppm, 1.218001.21800 ppm, 1.201001.20100 ppm.

Use the Grubbs test to determine if one of these values is an outlier. What is the value of ?calcGcalc?

?calc=Gcalc=

Should this potential outlier be rejected with 95% confidence? Critical values of ?G can be found in this table.

The potential outlier should be rejected.

The potential outlier should be kept.

Based on the outcome of the Grubbs test, calculate the mean (?¯x¯), standard deviation (?s), and 99% confidence interval for the results. A list of ?t values can be found in the Student's ?t table.

?¯=x¯=

ppm

?=s=

ppm

confidence interval:?¯±x¯±

ppm

Answer 1

Grubbs test:

Sample size, n =8

Mean, =(1. 21200 + 1.21200 +........ +1.20100)/8 =1.279125

Standard deviation, std = =0.201263

Suspected outlier =1.77700

Calculated Grubbs test value =G_calc = |Suspected outlier - mean|/Std =|1.77700 - 1.279125|/0.201263 =2.474

At n =8 and at 95% confidence level (5% significance level), the critical value (table value) of Grubbs test is: G_crit =2.032

Conclusion:

2.474 > 2.032

Since G_calc > G_crit, we reject the outlier. Thus, the potential outlier, 1.77700 should be rejected.

Now,

After removing 1.77700 from the data set,

the sample size of new data set is, n =7

Sample mean, =1.208

Sample std.deviation, s =0.006532

Standard Error, SE =s/ =0.006532/ =0.002469

Significance level =1 - 99% =1 - 0.99 =0.01 =1%

At 1% significance level, at n-1 =6 degrees of freedom, for a two-tailed case, the critical value of t is: t_crit =3.70743

Margin of Error, MoE =SE*t_crit =0.002469*3.70743 =0.009154

Lower bound = - MoE =1.208 - 0.009154 =1.19885

Upper bound = + MoE =1.208 + 0.009154 =1.21715

Thus, 99% confidence interval for the population mean, =(1.19885, 1.21715).