Question

In an effort to characterize the New Guinea crocodile (Crocodylus novaeguineae), measurements were taken of the dorsal cranial length (mm) (the length of the skull from the tip of the nose to the back of the cranial cap, denoted DCL) and the total length (cm) (denoted TL) of 50 harvested adult males.

DataTL DCL Observation 130 169 1 102 154 2 126 160 3 230 290 4 115 151 5 150 209 6 259 344 7 130 183 8 110 153 9 130 183 10 185 237 11 215 288 12 129 187 13 149 189 14 156 203 15 100 143 16 224 294 17 234 318 18 162 229 19 217 299 20 206 283 21 144 198 22 146 203 23 166 229 24 203 275 25 205 266 26 252 350 27 238 318 28 250 330 29 255 351 30 120 169 31 250 332 32 238 307 33

157 205 34 159 216 35 202 261 36 177 237 37 221 288 38 224 294 39 167 232 40 240 316 41 207 268 42 192 242 43 180 248 44 165 226 45 197 267 46 113 162 47 131 183 48 162 234 49 246 310 50

Question. (8). Is there a significant difference between DCL and TL measurements at the alpha=0.05 level? Report the null and alternative hypothesis. Complete the test and state your conclusion.

Answer 1

We need to test if there are differences between DCL and TL .

Since the same 50 adult males have been tested once for measuring DCL and then TL we need to apply for the Paired t-test here.

So the null hypothesis is H0: µD=0 (no diff) vs H1:µD ≠ 0 (significant diff)

where µD is the mean of the difference.of the two populations.

The test statistic is

t=((√n*dbar)}/sd

where dbar is the mean of the difference.of the two samples.

and sd is the standard deviation of the difference.of the two samples

We first find the difference di=xi-yi

where xi denotes DCL

and yi denotes TL

Table for calculation:

Observations	TL(yi)	DCL (xi)	di=xi-yi
1	102	154	52
2	126	160	34
3	230	290	60
4	115	151	36
5	150	209	59
6	259	344	85
7	130	183	53
8	110	153	43
9	130	183	53
10	185	237	52
11	215	288	73
12	129	187	58
13	149	189	40
14	156	203	47
15	100	143	43
16	224	294	70
17	234	318	84
18	162	229	67
19	217	299	82
20	206	283	77
21	144	198	54
22	146	203	57
23	166	229	63
24	203	275	72
25	205	266	61
26	252	350	98
27	238	318	80
28	250	330	80
29	255	351	96
30	120	169	49
31	250	332	82
32	238	307	69
33	157	205	48
34	159	216	57
35	202	261	59
36	177	237	60
37	221	288	67
38	224	294	70
39	167	232	65
40	240	316	76
41	207	268	61
42	192	242	50
43	180	248	68
44	165	226	61
45	197	267	70
46	113	162	49
47	131	183	52
48	162	234	72
49	246	310	64
50	130	169	39
Total			3117

dbar=3117/50=62.34

sd=√{(di-dbar)2/(n-1) =14.7836

n=50

The test statistic is

t=((√n*dbar)}/sd=(√50*62.34)/14.7836=29.8175~t distribution with degrees of freedom =n-1=50-1=49

The p-value for t=29.8175 and degrees of freedom 49 is very small < 0.00001

Hence, p-value is < α=0.05

Thus we reject the null hypothesis at α=0.05 and conclude that there are significant difference between DCL and TL .