Solutions to Mathematical Problems

Solutions
Q1)
x y X -
X (X - X )2 Y - Y (Y - Y )2 (X - X ) * (Y - Y )
5 20 -0.5 0.25 1.5 2.25 -0.75
3 23 -2.5 6.25 4.5 20.25 -11.25
7 15 1.5 2.25 -3.5 12.25 -5.25
9 11 3.5 12.25 -7.5 56.25 -26.25
2 27 -3.5 12.25 8.5 72.25 -29.75
4 21 -1.5 2.25 2.5 6.25 -3.75
6 17 0.5 0.25 -1.5 2.25 -0.75
8 14 2.5 6.25 -4.5 20.25 -11.25
∑ X=¿ ¿
44
∑ Y =¿ ¿
148 ∑ (X −X )2=¿ ¿
42 ∑ (Y −Y )2=¿ 192 ¿∑ ( X−X )∗( Y −Y )=−89
Mean (
X ¿=¿
5.5
Mean (
X ¿=¿
18.5
a)
Mean ( X ¿= (5+3+7+9+2+4+6+8)/ 8 = 5.5
Mean (Y ¿= (20+23+15+11+27+21+17+14)/8 = 18.5
∑ (X-X ) * (Y- Y ) = -89
COV (X, Y) = ∑ (X-Mean X)*( Y-Mean Y)/ N-1 = -89 / 7 = -12.71
b)
It means that the two variables have an inverse relationship. Higher values of one are associated
with lower values of the other
c)
Sx = ∑ (X - X )2/ N-1
Sx = 42/ 7 = 6
Sy = ∑ (Y - Y )2/ N-1 = 192 / 7 = 27.43

We got Sy = 6, Sy = 27.43
Cov(x, y) = -12.71
r(x, y) = Cov(x, y) / Sx * Sy
r(x, y) = -12.71 / 6*27.43 = -0.077
d)
Since the value of r(x, y) is negative so both X and y are negatively related with each other
Q2)
a) H0: P = 0.10
Ha: P ≠ 0.10
b) ^P = 81/900 = 0.09
Z =
p− ^p
√ p ( 1− p )
n
=
0.09−0.10
√ 0.10 ( 1−0.10 )
900
= -1
The P-value = 2 * P(Z < -1)
= 2 * 0.1587
= 0.3174
Since The P-value > ∝ = 0.05, so we should not reject H0.
So at 5% significance level there is enough evidence to infer that the competitor is correct.
c) For 95% confidence interval, the critical value z* = 1.96
The 95% confidence interval for population proportion is
^P +/- z* * √ P ( 1−P )
n

= 0.09 +/- 1.96 * √ 0.10 ( 1−0.10 )
900
= 0.09 +/- 0.0196
= 0.0704, 0.1096
d)
The values within the intervals of a 95% confidence will be considered acceptable, while values
that will be outside the intervals are always rejected. Therefore for the null hypothesis P = 0.10 is
contained in the 95% interval 0.0704 to 0.1096, this means that the null hypothesis will be
accepted at 0.05 level.
Q3)
(a) using excel to solve for the Measure of central tendency
For Mean use
= AVERAGE (A2:A31)
For median use
=MEDIAN (A2: A31)
For Mode use
=MODE (A2: A31).