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Assume we have a group of 10 rats daily injected with 50 µg Pb/kg b. wt. At the end of experiment, the Pb concentrations in the liver and kidney were measured and tabulated as mean ± standard error in the following table:Pb content Mean Standard error
liver (µg/g dry tissue)kidney (µg/g dry tissue)
6080
0.20.8
Is there any significant difference between the liver and kidneys in the levels of accumulated Pb at confidence level 95%?
Revision
2
So we want to test the null hypothesis H0: σ22 = σ1
2 against the alternate hypothesis HA: σ2
2 ≠ σ12 (2-tailed)
Solution:
∵𝑭 𝒄𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒆𝒅=𝒔𝟏𝟐
𝒔𝟐𝟐¿𝟐 .𝟓𝟑❑
𝟐
𝟎 .𝟔❑𝟐 ¿𝟏𝟕 .𝟕𝟖
F0.025(9,9) = 4.03
d.f.= 10 – 1 = 9
In this case, Fcalc (17.78) > Ftabulated (4.03), so we reject H0 that the two standard deviations are unequal, so P < 0.05
Does the type of diet significantly affected the body weight of mice at confidence levels of 95% and 99 %?
Three diets (I, II, III) for mice were tested for differences in body weight (in grams) after a specified period of time. The results are recorded in the following table:
Diet Body weight (g)
I 20 30 20 30
II 10 20 30 10
III 50 40 50 30
Compare between group I and II.
𝑯𝟎 :𝝁𝟏=𝝁𝟐=……….=𝝁𝒌
𝑯 𝑨 :𝒂𝒕 𝒍𝒆𝒂𝒔𝒕 𝒐𝒏𝒆𝒑𝒂𝒊𝒓 𝒐𝒇 𝝁′ 𝒔𝒓𝒆𝒏𝒐𝒕𝒆𝒒𝒖𝒂𝒍 .
Solution
Diet Body weight (g) mean size S
I 20 30 20 30 25 4 5
II 10 20 30 10 17.5 4 8.29
III 50 40 50 30 42.5 4 7.98
For I, σ2=25
For II, σ2=68.75
Source SS df MS F P
Between (Factor)
SSB dfB MSB
MSB/MSW
Within (Error)
SSW dfW MSW
Total SST
𝐗𝐠=𝟒 (𝟐𝟓 )+𝟒 (𝟏𝟕 .𝟓 )+𝟒 (𝟒𝟐 .𝟓 )
𝟒+𝟒+𝟒=𝟐𝟖 .𝟑𝟑
𝐒𝐒𝑩=𝐧𝟏 (𝐗𝟏−𝐗𝐠 )𝟐+𝐧𝟐 (𝐗𝟐−𝐗𝐠 )𝟐+𝐧𝟑 (𝐗𝟑−𝐗𝐠 )𝟐
= 1316.66dfB= h-1= 3-1= 2
MSB= SSB/dfB= 658.33
𝑺𝐒 (𝐖 )=𝒅𝒇 𝟏∗𝑺𝟏𝟐+𝒅𝒇 𝟐∗𝑺𝟐
𝟐+…+𝒅𝒇 𝒌∗𝑺𝒌𝟐
= 472.5dfW= N-h= 12-3= 9
MSW= SSW/dfW= 52.5
Source SS df MS FCal P
Between (Factor)
1316.66 2 658.33 12.54 <0.01
Within (Error)
472.5 9 52.5
Total 1789.16 11
F0.05 (2, 9)= 4.26F0.01 (2, 9)= 8.02
f-distribution Table
Tukey’s HSD (Honestly significance difference) Post-hoc test
𝑸=𝒒(𝒈 ,𝑵 −𝒈 ,∝ ) √𝑴𝑺𝑾𝒏
The critical value for comparison between two averages
Sample size /group
Number of groups = number of meansTotal number of Samples
Critical q value(tabulated)
Tukey’s HSD (Honestly significance difference) Post-hoc test
𝑸=𝒒(𝟑 ,𝟏𝟐−𝟑,𝟎 .𝟎𝟓 )√𝟓𝟐 .𝟓𝟒Q= 3.95 (3.62)=14.31
= 17.5, = 25, = 42.5
-= 7.5 <Q (14.31) insignificant
N (total sample size)- g (number of groups) g (3)12 – 3= 9
The data below represent the levels of blood glucose before and after injection with a certain herbal extract.
Experimental conditions Blood glucose levels (mg/dl)
Before 2, 3, 3, 2, 4, 2after 9, 8, 9, 8, 8, 7
Did the herbal extract cause increased blood glucose levels??
After (X2)
0
0
3
20
0
0
Before (X1) After (X2)
2 9
3 8
3 9
2 8
4 8
2 7
D
-7
-5
-6
-6
-4
-5
D2
49
25
36
36
16
25
𝒕=∑ 𝑫
√𝒏∑ 𝑫𝟐− (∑ 𝑫 )𝟐
𝒏−𝟏
𝑫=𝑿𝟐−𝑿𝟏 d.f. = n - 1
-33 187
=12.84
ttabulated at d.f. 5 = 2.015
𝑯𝟎 :𝝁=𝝁𝟎
𝑯𝟏 :𝝁<𝝁𝟎 ,
tCal (12.84) > ttabulated (2.015)Significant, P<0.05
In an experiment to study the effect of pH value on the hepatic Cd content, the data below were recorded. Test the claim that Cd content at pH 8 is significantly higher than at pH 5?
pH Hepatic Cd content (mg /kg b. wt.)5 4, 8, 8, 30, 10, 12, 128 4, 7, 10, 11, 14, 16,16, 21, 23, 25, 26
𝑯𝟎 :𝝁=𝝁𝟎
𝑯𝟏 :𝝁>𝝁𝟎 ,
pH5 (X1) pH8 (X2)
4 4
8 7
8 10
30 11
10 14
12 16
12 16
21
23
25
26
𝐒𝐒𝟏=∑𝐗𝟏𝟐−
(𝐗𝟏 )𝟐
𝐧𝟏
(X1)2
16
64
64
900
100
144
144
(X2)2
16
49
100
121
196
256
256
441
529
625
676
84 173 1432 3265
𝐒𝐒𝟏=𝟏𝟒𝟑𝟐−(𝟖𝟒)𝟐
𝟕=𝟒𝟐𝟒
𝐒𝐒𝟐=𝟑𝟐𝟔𝟓−(𝟏𝟕𝟑)𝟐
𝟏𝟏=𝟓𝟒𝟒 .𝟐
=12 =15.73
𝒕𝐂𝐚𝐥=𝐗𝟏−𝐗𝟐
√( 𝐒𝐒𝟏+𝐒𝐒𝟐
(𝐧𝟏+𝐧𝟐)−𝟐 )( 𝟏𝐧𝟏
+ 𝟏𝐧𝟐
)
17
d.f. = n1 + n2 - 2 = 7+11 -2= 16
=
= 1.001 ttabulated at d.f. (7+11-2) = 1.746
tCal (1.001) < ttabulated (1.746)
Insignificant, P>0.05
t-distribution Table
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