The PGA Players Championship was held at the Sedgefield Country Club in Greensboro, North Carolina, August 11–17, 2008. Shown here are first-round and second-round scores for a random sample of 11 golfers. Use α = .05 to determine whether the first- and second-round median scores for golfers in the Players Championship differed significantly. What is your conclusion? Golfer 1st Round 2nd Round Marvin Laird 63 74 Jimmy Walker 70 73 Kevin Chappell 72 70 Kevin Duke 65 71 Andrew Buckle 70 74 Paul Claxton 69 73 Larry Mize 72 71 Chris Riley 68 70 Bubba Watson 70 68 Carlos Franco 71 71 Richard Johnson 72 69

Golfer	Round		Absolute			Signed Ranks
Golfer	1	2	Difference	Difference	Rank	Negative	Positive
Marvin Laird	63	74	–11	11	10	–10
Jimmy Walker	70	73	–3	3	5.5	–5.5
Kevin Chappell	72	70	2	2	3		3
Kevin Duke	65	71	–6	6	9	–9
Andrew Buckle	70	74	–4	4	7.5	–7.5
Paul Claxton	69	73	–4	4	7.5	–7.5
Larry Mize	72	71	1	1	1		1
Chris Riley	68	70	–2	2	3	–3
Bubba Watson	70	68	2	2	3		3
Carlos Franco	71	71	0	0
Richard Johnson	72	69	3	3	5.5		5.5

The sum of the positive ranks can be denoted as T+. The Wilcoxon signed-rank test is conducted by using T+ as the test statistics.

Sum of the positive signed ranks is,

T+=1.5+7+3+3+3+5.5=12.5

It is known that if the sample size is greater than equal to 10, the approximation distribution of Wilcoxon signed rank test is Normal distribution.

Here, the difference for the golfer Carlos Franco is zero. Thus, the sample size becomes 10, which is equal to10. Thus, the distribution of T+ can be approximated as normal.

Mean, μT+=n(n+1)4

Substitute n=10 in mean of Wilcoxon signed-rank test and it is found that,

μT+=10(10+1)4=1104=27.5

Thus, the mean is 27

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Golfer	1st Round	2nd Round
Marvin Laird	63	74
Jimmy Walker	70	73
Kevin Chappell	72	70
Kevin Duke	65	71
Andrew Buckle	70	74
Paul Claxton	69	73
Larry Mize	72	71
Chris Riley	68	70
Bubba Watson	70	68
Carlos Franco	71	71
Richard Johnson	72	69