Statistics Homework: Detailed Solutions to Statistical Problems

This document presents a complete solution to a statistics homework assignment, addressing a range of problems. The solution includes detailed calculations and explanations for each question. The assignment covers topics such as finding critical points of functions, boundary calculations, derivative analysis, integral calculus, probability, and statistical measures like mean, standard deviation, and confidence intervals. Various statistical concepts are applied to solve each problem, providing a comprehensive guide for students studying statistics. The document demonstrates the application of formulas and methods to arrive at the solutions, making it a valuable resource for understanding and learning statistical concepts.

Running head: Statistics 1
Statistics
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Statistics 2
Question 1
F(x) = ax4-bx2
F’(x) = 4ax3-2bx
F’(x) =0 = 4ax3 -2bx
X (2ax2-b) = 0
X = 0
2ax2 = b
But a =67.5, b = 54.4
135x2 = 54.4
X2 = 0.40
X = 0.63
F(0) = 0
F(0.63) = 67.5 (0.63)4 – 54.4 (0.63)2
= 10.63 – 21.59
= -10.96
Critical points = (0, 0) (0.63, -10.96)
Greatest absolute value = 0
Question 2
Boundaries (86, 86.2)
Nax(n/m)-1 + -kbx-k-11
= 16x - 372x-7
=16(86.2) – 372(86.2)-7 – 16(86) – 372 (86)-7
= 1379.2- 1376
= 3.2
Question 3
F(x) = a/xb
F(x) = a.x-b
F’(x) = -bax-b-1
F’’(x) = b2+baxb+1

Statistics 3
But a=23, b=89
89(90+1)23x89+1
= 8010.23x90
= 184,230x90
c-d = 184,140
Question 4
∫
1
3
f ( x ) dx−kxdx = a
∫
1
3
b−kxdx =13
∫
1
3
7−kxdx =13
∫
1
3
−6=k ∫ xdx
-6x = kx2/2
-18 = 4.5k
K=-4
-6 = 0.5k
K =-12
-4+12 = 8
K =8
Question 5
Dy/dx = 2x +q/5x2
Gradient at turning point, derivative =0
2x +q/5x2 =0
= 5x2 – q = 0

Statistics 4
5x2 =q
Q = 281.25
Question 6
∫
a
b
x ❑c+d
x dx
= ∫
a
b
❑ xc/x + d/x dx
= ∫
a
b
❑ xc+1/x (c+1) + d2/2x dx
But a=2.4, b=10.5, c=2 and d=30
= x3/3x + 90/2x
= 10.53/(10.5) + 90/2(10.5) – 10.52.4/3(2.4) + 90/2(2.4)
= 36.75 + 4.286 – 39.22 + 18.75
= 41.036 – 57.97
= -16.93
Question 7
∫
❑
❑ ax +b
x ❑2 −9 x+20 dx
= ax2/2 +bx+c – x3/3-9x2/2+20x +c
= 10x2+4.5x2-69x-20x-x3/3
=14.5x2-89x-3lnx3+c
=14.50
Question 8
∫
❑
❑ ax +b
x ❑2 +5 x−14 dx
ax ❑2
2 +bx+c
x 3
3 + 5 x 2
2 −14 x +c
dx
=38.5x2+bx+c-(3lnx3-2.5x2-14x+c)
=41x2+bx+14x-3lnx

Statistics 5
= 41
Question 9
Y=ax2-bx
Y = cx-dx2
ax2-bx=cx-dx2
ax2+dx2=cx-bx
= x2(a+d)=x(c-b)
=137.8x2=-18.8x
=x(137.8x-18.8)=0
X=0 0r x=137.8/18.8
=-7.33
A= ∫
−7.33
0
( x−dx 2 )−(ax 2−bx)
= -7.332/2 – 84.3 (-7.33)3/3- 53.5 (-7.33)3 – 54.8 (-7.33)2/2
= 26.86 + 11066.70
= 19,589.08
Area = 0-19,589.08
=-19,589.08 square units
Question 10
a= Vf-Vi/t
v= D/T
48 = 344m/t
T = 7.17 seconds
a= 48/7.17
= 6.60 m/s
Question 11
In a 3 fair dice there are 6x6x6 =216 possible outcomes
Chances of appearing on 1st die = 1/6
There are six ways to get 555 only

Statistics 6
Hence, 6/216
=0.03
Question 12
P(1 tripple) = 11/36
But 400 rolls
=1/36 x 400
=11.1111
Question 13
P(choosing an integer between 1 and 56) = 1/56
Possible outcomes = 112
P(only 2 men choose same) =56/112
=0.500
Question 14
Probability of choosing same number = 1/49
P(choosing different number) = 1-1/49
5 people = 1-1/49*5
=4.8979
Question 15
Possible outcomes 25 = 32
P(all outcomes same) = 1/16
P(not same) = 1-1/16 = 15/16
=0.9375
Question 16

Statistics 7
x x-xbar
x-xbar
sqrd
16 -10.2857 105.7959
11 -15.2857 233.6531
23 -3.28571 10.79592
29
2.71428
6 7.367347
44
17.7142
9 313.7959
34
7.71428
6 59.5102
27
0.71428
6 0.510204
184 731.4286
26.2857
1
=√ 731.4286/7
= 11.04
Question 17
x f x-xbar
x-xbar
sqrd
f(x-xbar
sqrd)
18 13 -18 324 4212
38 29 2 4 116
52 36 16 256 9216
total 108 78 584 13544
mean 36
= √13544/78
=13.18
Question 18
Z values for CI of 95% = 1.96
= 31.71+- 1.96 *12.39/√71

Statistics 8
= 31.71 + 2.88 or 31.71- 2.88
Upper limit = 34.59
Question 19
Z values for 99% CI = 2.58
=21.04+- 2.58 *2.56/√307
= 21.04+- 0.37
Lower limit
=21.04-0.37
=20.66
Question 20
= 4.4+- 2.58 * 0.37/√1126
= 4.4 +- 0.03
Lower limit
= 4.37