Question

Numerator Degrees of Freedom de df, 2 3 4 5 7 8 9 1 2 3 4 5 6 7 3 Denominator Degrees of Freedom 10 11 12 13 14 15 16 17 161.45 18.513 10.128 7.7086 6.8079 59874 5.5914 5.3177 5.1174 4.9646 4.8443 4.7472 4.6672 4.6001 4.5431 449-40 4.4513 4.4139 4.3807 4.3512 4.3248 4.3009 4.2793 4.2597 4.2417 4.2252 4.2100 4.1960 4.18.30 4.1709 4.0847 4.0012 3.9201 3.8415 199.50 19.000 9.5521 9.9443 5.7861 5.1433 4.7374 4.4590 4.2565 4.1020 3.9823 3.8853 3.8056 3.7389 3.6823 3.6137 3.5915 3.5546 3.5219 3.4928 3.4668 3.4434 3.4221 3.4028 215.71 19.164 9.2766 6.5914 5.4095 4.7571 4.3468 4.0662 3.8625 3.7083 3.5874 3.4901 3.4105 3.1419 3.2874 3.2189 3.1968 3.1599 3.1274 3.0984 3.0725 3.0491 3.0280 3.0088 2.9912 2.9752 2.9604 2.9467 2.9340 2.9223 28382 2.7581 2,6802 2.6049 224.58 19.247 9.1172 6.3882 5.1922 4.5337 4.1203 3.8379 3.6331 3.4780 3.3567 3.2592 3.1791 3.1122 3.0556 3.0069 29647 2.9277 2.8951 2.8661 2.8401 2.8167 2.7955 2.7763 2.7587 2.7426 2.7278 2.7141 2.7014 2.6896 2.600 2.5252 2.4472 2.3719 230.16 19.296 9.0135 6.2561 5.0803 4.3874 3.9715 3.6875 34817 3.3256 3.2039 3.1059 3.0254 2.9582 2.9013 28524 2.8100 2.7729 2.7401 2.7109 2.4848 2.6613 2.6400 2.4207 2.30 2.5868 2.5719 2:55 25454 2:5316 2.495 2 1683 2.2899 22141 233.99 19.310 8.9406 6.1631 4.9503 4.2839 3.8660 3.5806 3.3738 3.2172 3.0946 2.9961 2.9153 2.8477 2.7905 2.7413 26987 2.6613 2.6283 2.5990 2.5727 2.5491 2.5277 2.5082 2.4904 24741 2:4591 2 453 24124 24205 21159 22541 2.750 2095 236.77 238.38 19.353 19371 8.8867 8.8452 6.0942 6.0410 4.8759 48183 4.2067 4.1465 3.78703.7257 3.5005 3.4381 3.2927 3.2296 3.1355 3.0712 3.0123 2.9480 2.914 2.8486 2.8321 2.7669 2.7642 2.6987 2.7066 2.6408 2.6572 2.5911 26143 2.540 2.5767 2.5102 2.5435 2.4768 2.5140 2.4471 2.4876 2.4205 2.4638 23965 24422 2.3745 2.4226 23551 2.4047 2.3371 2.3883 2.3205 2372 2003 23593 2.2913 2.3463 2.2783 2.1143 2.2662 22490 2.1802 20970 ZORAN 2.0164 2009 1 94 240.54 19.385 8.8123 6.9988 4.7725 4.0990 3.6767 3.3881 3.1789 3.0204 2.8962 2.7964 2.7144 2.4458 2.5876 2.5377 2.4943 24563 24227 2.3928 2 3660 23419 23201 2.3002 2.2821 2.2655 22501 22360 2 2229 2 2107 2.1240 2.0401 1.9588 18799 20 21 22 23 24 25 26 27 28 29 0 3.3620 3.3541 3.3404 3.3277 31S 3.2317 3.1504 3.0718 2.9957 40 120

Answer 1

There are three treatments.

The overall means for all 12 observations, that is grand mean

$\bar \bar x = \frac{Sum \ of \ all \ observations}{total \ observations} = \frac{372}{12} =31$

Means for each tratment

1: Valerian Root, 2: Melatonin and 3: 5- HTP

$\bar x_1 = \frac{Sum \ of \ all \ observations \ of \ valrian \ root}{total \ observations} = \frac{118}{4} =29.5$

141 12 Sum of all observations of Melatonin total observations = 35.25 4.

$\bar x_3 = \frac{Sum \ of \ all \ observations \ of \ 5-HTP}{total \ observations} = \frac{113}{4} =28.25$

The formula to find the total sum of squares

$SST = \sum \sum (x_{ij}-\bar \bar x)^2$

Subtact overall mean 31 from each 12 observations then take squares and add all of them to get the total sum of squares.

$SST = \sum \sum (x_{ij}-\bar \bar x)^2 = 174$

The formula for within sum of squares,

$SSW = \sum_{j}\sum_{i}(x_{ij}-\bar x_j)^2$

Subtract each treatment mean from the 4 observations of that same treatment, then take squares. Repeat same for the rest two treatments and then add all the squares.

The first treatment is 29.5 so subtract it from the observations 29, 32, 27 and 30 then take squares. Similarly subtract the means 35.25 and 28.25 from their corresponding observations and take square of them.

After that add all the 12 squares like 4 for first treatment, 4 for second and 4 for last treatment.

$SSW = \sum_{j}\sum_{i}(x_{ij}-\bar x_j)^2= 62.5$

Between sum of squares

SSB = SST - SSE = 174 - 62.5 = 111.5

Degrees of freedom

Degrees of freedom for between = k - 1 = 3 - 1 = 2

Degrees of freedom for within = n - k = 12 - 3 = 9

Degrees of freedom for total = n - 1 = 12 - 1 = 11

Mean square sums

Between mean sum of square(MSB)

$MSB = \frac{SSB}{DF \ for \ between} = \frac{111.5}{2} = 55.75$

Within mean sum of square (MSW)

$MSW = \frac{SSW}{DF \ for \ within} = \frac{62.5}{9} = 6.944$

F test statistics

$F = \frac{MSB}{MSW} = \frac{55.75}{6.944} = 8.028$

F test value = 8.028

The ANOVA table is,

Source of variation	Sum of squares	DF	Mean square	F
Between groups	111.5	2	55.75	8.028
Within groups	62.5	9	6.944
Total	174	11

The degrees of freedm for between and within groups are correct. Also the critical value at 0.05 is correct.