1. 3+5+7+......+(2n+1)=n (n+2) for N 1 Solution 3+5+7+......+(2n+1)=n (n+2) for N 1 base case when n = 1 3 = 1*(1+2) = 3 this is true thus assume that given is true for n =k thus 3+5+7+......+(2k+1)=k (k+2) for N 1 thus we have to prove that this is true for n = k+1 too now add 2(k+1)+1 = 2k+3 both sides 3+5+7+......+(2k+1)+ 2k+3= k(k+2) +(2k+3) k(k+2) +(2k+3) = k^2+2k+2k+3 = k^2+4k+3 = (k+1)(k+3) which is true for n = k+1 thus given is true according to mathematical induction :).

1. 1. 4.The following data are from a two-factor study examining the effects of three treatment
conditions on males and females.
a. Use an ANOVA with ? = .05 for all tests to evaluate the significance of the main effects and
the interaction.
b. Plot the cell means and describe the interaction, if it is significant.
I
II
III
Male
1,
2,
6,
M=3
7,
2,
9,
M=6
9,
11,
7,
M=9,
Female
3,
1,
5,
M=3
10,
11,
15,
M=12
16,
18,
11,
M=15

2. I
II
III
Male
1,
2,
6,
M=3
7,
2,
9,
M=6
9,
11,
7,
M=9,
Female
3,
1,
5,
M=3
10,
11,
15,
M=12
16,
18,
11,
M=15
Solution
SS df MS F p Between: 54.000 2 27.000 3.385 0.104 Within: 47.859 6 7.977 Total:
101.859 8 For male For female SS df MS F p Between: 234.000 2 117.000 14.668 0.005 Within:

3. 47.859 6 7.977 Total: 281.859 8 For alpha =0.05 we have 0.842 as answer