2. Independent random samples from two regions in the same area gave the followi

Question

2. Independent random samples from two regions in the same area gave the following chemical

measurements (ppm). Assume the population distributions of the chemical are mound-shaped and

symmetric for these two regions.

Region I: x 1 ; n1 = 12

761 620 990 425 746 403 734 429 469 518 934 635

Region II: x 2 ; n2 = 16

413 480 605 894 496 1080 998 363 339 751 378 336 1016 741 1113 344

Let 1 = 639 be the population mean and s 1 = 199 be the population standard deviation for x 1 . Let

2 = 647 be the population mean and s 2 = 293 be the population standard deviation for x 2 .

Determine and examine the 90% confidence interval for 1 2 . Does the interval consist of

numbers that are all positive? all negative? or different signs? At the 90% level of confidence, is

one region more interesting that the other from a geochemical perspective?

A) The interval contains only positive numbers. We cannot say at the required

confidence level that one region is more interesting than the other.

B) The interval contains only negative numbers. We cannot say at the required

confidence level that one region is more interesting than the other.

C) The interval contains both positive and negative numbers. We cannot say at the

required confidence level that one region is more interesting than the other.

D) The interval contains only negative numbers. We can say at the required

confidence level that one region is more interesting than the other.

E) The interval contains only positive numbers. We can say at the required

confidence level that one region is more interesting than the other.

Explanation / Answer

The  standard error (SE) of the sampling distribution of difference in means is

SE = sqrt[ (s12/n1) + (s22/n2) ]

where s1 is the  standard deviation of region 1, s2 is the standard deviation of region 2, n1 is the size of region 1, and n2 is the size of region 2.

SE = sqrt[ (1992/12) + (2932/16) ] = 93.0894

Z value for 90% confidence is 1.64

Difference in means 1 2 = 639 - 647 = -8

90% confidence interval for 1 2 is,

(-8 - 1.64 * 93.0894, -8 + 1.64 * 93.0894)

= (-160.6666, 144.6666)

The interval contains both positive and negative numbers. So, the answer is the interval consist of numbers that are of different signs.

As, the 90% confidence interval contains both positive and negative numbers, the correct option is

C) The interval contains both positive and negative numbers. We cannot say at the required confidence level that one region is more interesting than the other.

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