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Probability & Statistics

A magazine reported on a study of the reliability of a commercial kit to test for arsenic in groundwater. The field kit was used to test a sample of 20 groundwater wells in a country. In addition to the arsenic level (micrograms per liter) the latitude (degrees) longitude (degrees) and depth (feet) of each well was measured.

 

 Table:
 
Well_ID     Latitude    Longitude         Dept       Arsenic
1               23.7714   90.6221             30          326
 
2               23.7764   90.6492             60           326

3               23.7832   90.6321            125          158
 
4               23.7944   90.6386             30           223
5               23.7644   90.6224             100           16
6               23.7916   90.6228             100           16
7               23.7674   90.6225               45          201
8               23.7694   90.6176               90          320
9               23.7732   90.6147               75          627
10             23.7894   90.6482             125          216
11             23.7852   90.6215               75           13
12             23.7856   90.6207               45          143
13             23.7745   90.6496               90          169
14             23.7479   90.6387               60          269
15             23.7734   90.6487               90          281
16             23.7681   90.6463               90          326
17             23.7754   90.6452               90          631
18             23.7881   90.6489             125            43
19             23.7872   90.6218               30          189
20             23.7818   90.6243              125          260
 
 
 
 
 
 
Fit the model to the data using the method of least squares. Let x1 be the latitudes, x2 be the longitudes, x3 be the depths

Answer. y= (?) +(?) x1 (?) x2+(?) x3  Round the constant term and the coefficients for x1 and x2. Round the coefficient for x3 to the nearest hundredth as needed
 

f.Conduct a test of overall model utility at a=0.01. what are the null and alternative hypothesis?

Answer.            H0: b1 =b2=b3=0 

                        Ha: At least one b1 ≠ 0 

What is the test statistic?

F=

What is the p-value?

p=
 
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1 Answer

This problem can be done by hand, of course, but it requires a lot of computations as you have a multiple regression. Are you allowed to use a calculator such as the TI-83 or higher graphing calculator? If you are, the answers are not too difficult to get.
 
1) Put the data in L1, L2, L3, L4   where L1 is arsenic, your dependent variable.
2) Press 2nd MATRIX then select MATH and select List>matr   + ENTER.
3) Put L1, L2, L3, L4 in the matrix as follows: List>matr(L1, L2, L3, L4, [D])
 
To get your regression Press PRGM, Select A2MULREG + ENTER + ENTER (in this way you select MULT REGRESSIO
 
You will be asked
How many ind var?  you will enter  3 + ENTER
Col. of var. 1 ?  2 + ENTER
Col of var. 2 ? 3 + ENTER
Col of var. 3 ? 4   + ENTER
 
This will produce the outputs that you need
 
 
Select QUIT + ENTER to exit
 
If you are a college student and you use other packages then your approach is going to be somewhat different but not too much.  I use R which is a free program of the highest standard to perform regressions.
 
You should let us know what you need to do (by hand, by computer, by calculator), what software you have available, etc..

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