Question

Wilcoxon signed-ranks test for method Services data of self-reported and measured his for d evidence to support the claim that there in med and measured heights of males aged 12-16. Use a =0.05 of Health and Human 12-16. Is there s om between sell reported 63 56 PS20556 742 65 Reported height 68 71 20 21 Measured height 679 699 BOA203 Differenced) Rank of Signed Rank 72 70.8 a. From the claim derive Ho and HD b. Fill up the Table above and find n, sum of positive ranks and negative ranks. Positive ranks sum Negative ranks sum= c. What is the T value? d. From the Table A-8 find critical T value, e. State conclusion about HO and Claim. Hint: compare T-value and critical T-value 3. You are given cholesterol levels of 13 men and 12 women as below. Use Wilcoxan Rank Sum Test a. Sort 25 numbers and rank them from smallest to largest, and fill up the ranks in the table. Male Rank 522 127 b. Find in and CR 740 9 230 316 590 126 C. Calculate test statistic z. 465 121 578 130 255 d. Test the claim that the two populations of men and women have different median cholesterol levels. Can you support the claim? Hint: find Ho and Hı, two tailed test? Critical values from Table A-2. Use traditional method with a +0.05. 250 ni= RIE

Answer 1

Reported Height	68	71	63	70	71	60	65	64	54	63	66	72
Measured Height	67.9	69.9	64.9	68.3	70.3	60.6	64.5	67.0	55.6	74.2	65	70.8
Difference(d)	0.1	1.1	-1.9	1.7	0.7	-0.6	0.5	-3	-1.6	-11.2	1	1.2
Rank(d)	1	6	10	9	4	3	2	11	8	12	5	7
Signed Rank	+1	+6	-10	+9	+4	-3	+2	-11	-8	-12	+5	+7

a)

Null and Alternative Hypotheses:

$H_0:$There is no difference in median between self reported heights and measured heights .

$H_1:$There is a difference in median between self reported heights and measured heights .

b)

Here n=12

The sum of positive ranks is:

\

w +=1+2+4+5+6+7+9 = 34

and the sum of negative ranks is:

W-= 3+8 + 10 + 11 + 12 = 44

c)

The test statistic is

T = min{W+,w-} = min{34, 44} = 34

d)

The critical value for the significance level
g= 0,05
provided, and the type of tail specified is
T* = 13
, and the null hypothesis is rejected if
T< 13
.

e)

Since in this case
T = 34 > 13
, there is not enough evidence to claim that the population median of differences is different than 0, at the
g= 0,05
significance level.

Thus we can conclude that there is a significant difference in median between self reported heights and measured heights.

3) Wilcoxon Rank-Sum Test

Male	Rank	Female	Rank
522	21	264	15
127	9	181	12
740	25	267	17
49	2	384	19
230	13	98	6
316	18	62	3
590	24	126	8
466	20	89	5
121	7	531	22
578	23	130	10
78	4	175	11
265	16	44	1
250	14
$n_1=$13 $R_1=$196		$n_2=$12 $R_2=$129

The sum of ranks for sample 1(Male) is:

Ri=2+4+7+9+13+14+16+18+ 20 +21+23+24+25 = 196

and the sum of ranks of sample 2(Female) is:

R2=1+3+5+6+8+10 + 11 + 12 + 15 + 17 + 19+22 = 129

Hence, the test statistic is  .

R= R1 = 196

The following null and alternative hypotheses need to be tested:

$H_0:$The two populations of men and woman have same median cholesterol levels .or $H_0​$ : Median (Difference) = 0

$H_1:$The two populations of men and woman have different median cholesterol levels .or $H_1$ ​: Median (Difference) $\neq$ 0

Observe that both sample sizes are greater than 10, then we can use normal approximation. The following z-statistic will be used.

The z-statistic is computed as follows:

R-nini+n2+1)/2 ning(nı + n2+1)/(12) 196 – 13(13+12+1)/2) 13 * 12(13+ 12+1)/12 = 1.469

Based on the information provided, the significance level is
g= 0,05
, and the critical value for a two-tailed test is .The rejection region for this two-tailed test is
R= 2: : > 1.96
.

Since it is observed that , it is then concluded that the null hypothesis is not rejected.

12 = 1.469<%= 1.96

Using the P-value approach: The p-value is
p=0.1419
, and since
p= 0.1419 > 0.05
, it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population median of differences is different than 0, at the 0.05 significance level.

Graphically:

p-value = 0.1419 0.35 0.30 0.25 0.20 0.15- 0.10 0.05 - 0.00 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0