Question

In an effort to compare vegetation growth in an area a cluster sample was done in 2006 at 8 locations each 1 square foot and mapped with GPS coordiantes.

The next year the researchers came back to the same area to see if vegetation had increased. They used the coordinates and mapped the same square foot units. The data is in the table below. Test to see if vegetation has increased in 2007 from the year previous.  

	2006	2007
Location A	124	132
Location B	35	56
Location C	21	12
Location D	245	270
Location E	151	180
Location F	78	81
Location G	95	11
Location H	82	96

Complete all 8 steps of a hypothesis test. You may skip making a histogram.  

Answer 1

• Step 1
  Under Null, we assume that the there is no increase i production in both the yeers.
  The Null hypotheis is set as:
  
  
  
• Step 2
  We are to test, if the production has increased in the following year.
  Hence, the alternative is given by
  
  
  
• Step 3
  We allow an utmost 5% Type I error. That is, we set,
  
  
  
• Step 4
  The data is given by:

  Location	2006	2007
  Location A	124	132
  Location B	35	56
  Location C	21	12
  Location D	245	270
  Location E	151	180
  Location F	78	81
  Location G	95	11
  Location H	82	96

• Step 5
  It is assumed that the underlying distribution is Normal.
  
  
  
• Step 6
  Since, the plots are correlated,
  we use the test statistic,
  
  where,
  
  
  We modify the table as

  Location	2006	2007	d
  Location A	124	132	8
  Location B	35	56	21
  Location C	21	12	-9
  Location D	245	270	25
  Location E	151	180	29
  Location F	78	81	3
  Location G	95	11	-84
  Location H	82	96	14

  
  Note, here, we test the hypothesis
  
  
  Calculations
  
  
  
  n = 8
  
  Hence,
  
  
  
  
• Step 7
  The test is rejected iff
  Now,
  
  
  Hence, the Null Hypothesis is rejected iff T > 1.895.
  
  
  
• Step 8
  It is observed that,
  
  
  Hence, we fail to reject the Null Hypothesis and conclude that, there is no increase in vegetation.
  
  
  
  

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