Assessment Task 2: QAB 105 Quantitative Analysis for Business

This document provides a comprehensive solution for QAB 105, a Quantitative Analysis for Business course assignment. The solution addresses key concepts including random sampling and calculating 95% confidence intervals. It delves into probability distributions, calculating various statistical values. Furthermore, the assignment solution tackles hypothesis testing, determining p-values and drawing conclusions based on significance levels. Finally, the document includes a detailed linear regression analysis, interpreting regression lines, slope coefficients, and assessing the strength of the relationship between variables using correlation coefficients and scatter plots, including references. The assignment is designed to help students understand and apply statistical methods in a business context.

QAB 105 Quantitative Analysis for Business
Assessment Task 2
Student Id
[Pick the date]

Question 1
Random sample = 200
Number of retailers sold products below the minimum price = 79
(a) A random sample is a population subset where every element has an equal chance or
probability of being selected. A random sample can be selected by ensuring that all the
retailers be given a unique number and then 200 numbers can be randomly selected from
the whole population of numbers through software to avoid any bias. The retailers
corresponding to the number selected would serve as part of the random sample (Hillier,
2006).
(b) 95% confidence limits for the proportion
Mean ¿ 79
200 =0.395
Z value for 95% confidence interval = 1.96
Upper limit of 95 %confidence limits=¿
¿ 0.395+ ( 1.96∗
√ 0.395 ( 1−0.395 )
200 )
¿ 0.395+ ( 1.96∗0.0345 )
¿ 0.4627
Limit limit of 95 % confidence limits
¿ 0.395−(1.96∗
√ 0.395 ( 1−0.395 )
200 )
¿ 0.395− ( 1.96∗0.0345 )
¿ 0.327
95% confidence interval [0.327 0.4627]
1

Question 2
Probability distribution
X -4 0 1 2
P(x) 0.2 0.3 0.4 0.1
a.) P ( X >0 )
The value of P ( X >0 )
¿ P ( X=1 ) + P ( X =2 )
¿ 0.4+ 0.1
¿ 0.5
b.) P ( X ≤0 )
The value of P ( X ≤0 )
¿ P ( X=0 ) +P ( X=−4 )
¿ 0.3+0.2
¿ 0.5
c.) P ( x ≤ X ≤1 )
The value of P ( x ≤ X ≤1 )
¿ P ( X=1 ) + P ( X < 0 ) + P ( X ←4 )
¿ 0.4+ 0.3+0.2
¿ 0.9
2

d.) P ( X=−2 )
The value of P ( X=−2 )
¿ 0
e.) P ( X=−4 )
The value of P ( X=−4 )
¿ 0.2
f.) P ( X <2 )
The value of P ( X <2 )
¿ P ( X=1 ) + P ( X =0 ) + P( X=−4)
¿ 0.4+ 0.3+0.2
¿ 0.9
Question 3
Mean value μ = $88950
Sample size n = 12
Mean valuation x= $92460
Standard deviation s = $5200
Significance level ∝=0.05
Prices are normally distribution.
Step 1:
Hypotheses are highlighted below:
3

H0 Nullhypothesis : μ=88950
H1 Alternative hypothesis :μ >88950
Step 2:
The value of t statistics is computed below:
t= x−μ
s
√n
t= 92460−88950
5200
√ 12
t=2.338
Step 3:
Degree of freedom = n-1 = 12 -1 = 11
Step 4:
For t statistics = 2.338 and degree of freedom = 11, the p value (one tailed test) is computed as
0.0196.
Step 5:
It is apparent that p value is lower than significance level (0.0196 <0.05). Hence, null hypothesis
would be rejected and alternative hypothesis would be accepted (Flick, 2015). Therefore, it
would be fair to conclude that mean prices of the Homes in Blacktown city is higher than
$88,950.
Question 4
Given price and cost output data
4

(a) Linear regression analysis
Dependent variable = Cost ($000’)
Independent variable = Units if Output (000’)
Regression line
Cost =21.25+(0.9615∗Unit of outputs)
Interpretation
 Intercept = 21.25
It represents the cost when the unit of outputs produced is zero.
 Slope coefficient = 0.9615
It represents that when there is one unit increase in the unit outputs then the cost would be
increased by a factor of 0.9615 (Hastie, Tibshirani and Friedman, 2011).
5

(b) Strength of relationship between costs and units of outputs
Correlation coefficient would be determined with the help of CORREL function.
Scatter plot
5 10 15 20 25 30 35 40 45 50 55
0
10
20
30
40
50
60
70
80
Scatter Plot
Units of outputs (000')
Cost ($000')
Based on the above scatterplot and the correlation coefficient, it is apparent that there is a strong
positive association between the given variables. The positive association is reflected from the
positive slope of the scatter plot and the sign of the correlation coefficient. The magnitude of the
correlation coefficient reflects the intensity of linear relationship (Hair et. al., 2015).
6

References
Flick, U. (2015). Introducing research methodology: A beginner's guide to doing a research
project, 4th ed., New York: Sage Publications.
Hair, J. F., Wolfinbarger, M., Money, A. H., Samouel, P., and Page, M. J. (2015). Essentials of
business research methods, 2nd ed., New York: Routledge.
Hastie, T., Tibshirani, R. and Friedman, J. (2011). The Elements of Statistical Learning, 4th ed.,
New York: Springer Publications.
Hillier, F. (2006), Introduction to Operations Research, 6th ed., New York: McGraw Hill
Publications.
7