7011GBS Quantitative Methods Assignment 2: Statistical Analysis, 2019

This document provides a comprehensive solution to the 7011GBS Quantitative Methods Assignment 2, which includes detailed answers to three questions. The first question focuses on hypothesis testing, exploring scenarios related to online retail profitability, e-gift card usage, and the comparison of appraiser means using t-tests. The second question delves into multiple regression analysis, examining the impact of parental and grandparental ages on an individual's longevity, and incorporates a smoker dummy variable. The analysis includes identifying significant slope coefficients and model comparisons. The third question involves multiple regression analysis to determine the relationship between the number of boxes, weight, and time required to unload trucks, using dummy variables for time of day. The solution includes interpreting slope coefficients, comparing simple and multiple regression models, and determining the significance of the time-of-day variable. The assignment utilizes statistical techniques to analyze real-world scenarios and interpret results.

Question 1
(a) Hypothesis testing
Whether the online retail store will be termed as profitable when the mean order is more than
$85. It is apparent that population standard deviation is unknown and thus, t test would be
used in place of z test.
Here, it is apparent that p value (0.013) is lesser than level of significance i.e. 0.05 and thus,
null hypothesis will be rejected and alternative hypothesis will be accepted. Therefore, the
online retail store will be termed as profitable when the mean order is more than $85.
(b) Hypothesis testing
The proportion of people who has got an e-gift card are higher than 20%.
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Here, it is apparent that p value (0.1807) is more than level of significance i.e. 0.05 and thus,
null hypothesis will not be rejected and alternative hypothesis will not be accepted.
Therefore, the proportion of people who has got an e-gift card are not higher than 20%.
(c) Hypothesis testing
Whether the two appraisers mean are statistically different or not.
Null hypothesis Ho : μApp raiser 1 − μAppraiser 2 = 0
Alternative hypothesis Ha: μAppraiser 1 − μAppraiser 2 ≠ 0
It is apparent that population standard deviation is unknown and therefore, the appropriate
test would be t test. Further, the two samples mean are independent and thus, two sample
independent t test would be used.
Two tailed hypothesis p value = 0.7414
Here, it is apparent that p value (0.7414) is more than level of significance i.e. 0.05 and thus,
null hypothesis will not be rejected and alternative hypothesis will not be accepted.
Therefore, the two appraisers mean are not statistically different.
Question 2
(a) Multiple regression analysis
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Regression equation
Longevity = 3.244 + ሺ0.451 ∗ Motherሻ+ ሺ0.411 ∗ Fatherሻ+ ሺ0.017 ∗ Gmothersሻ
+ (0.087 ∗ Gfathers)
(b) Slope coefficients
Mother: This implies that when the age of mother has increased by 1 year then the
corresponding age of the person would increase by 0.451 years.
Father: This implies that when the age of father has increased by 1 year then the
corresponding age of the person would increase by 0.411 years.
Gmother: This implies that when the age of grandmother has increased by 1 year then the
corresponding age of the person would increase by 0.017 years.
Gfather: This implies that when the age of grandfather has increased by 1 year then the
corresponding age of the person would increase by 0.087.
Hypothesis testing
Ho : Slope is insignificant.
HaSlope is significant.
Null hypothesis will be rejected when the p value of the slope coefficient is lower than
significance level (0.05).
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Slope coefficients Mother and Father are significant and grandmother and grandfather are
insignificant.
(c) The longevity of man for the given inputs needs to be determined.
Mother = 75,
Father =75,
Grandmothers = 77,
Grandmothers =73
Longevity = 3.244 + ሺ0.451 ∗ Motherሻ+ ሺ0.411 ∗ Fatherሻ+ ሺ0.017 ∗ Gmothersሻ
+ (0.087 ∗ Gfathers)
Longevity = 3.244 + ሺ0.451 ∗ 75ሻ+ ሺ0.411 ∗ 75ሻ+ ሺ0.017 ∗ 77ሻ+ (0.087 ∗ 73) = 75.5
(d) Multiple regression model
The main difference of the two regression models are listed below.
The slope coefficient or mother and father have reduced as the impact of their age on the age
of the child has reduced. Further, the p value of grandmother and grandfather has also
reduced which implies that these slope coefficient’s significance has improved.
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(e) Smoker dummy variable
It implies that when the persona is categorised as smoker then the age of that person would
decrease by 3.719 as compared with a non-smoker by keeping the rest variable as constant.
Hypothesis testing
Ho : Slope is insignificant.
HaSlope is significant.
Null hypothesis will be rejected when the p value of the slope coefficient is lower than
significance level (0.01). Here, the p value for slope coefficient (smoker) is zero which means
it is lower than significance level. Thus, sufficient evidence is present to reject null
hypothesis. Hence, the smoking has significant effect on length of life.
Question 3
(a) Multiple regression analysis
Time = -28.427 + 0.604*Boxes + 0.374*Weight
This implies that when the number of boxes has increased by 1 then the corresponding time
to unload box would increase by 0.604 minutes.
This implies that when the weight of box has increased by 100 kg then the corresponding
time to unload would increase by 0.374 minutes.
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(b) Simple regression
The above shown model would not be considered suitable because there is uniform difference
is present in the time of unloading the boxes between early and late afternoon and morning
and early afternoon. Further, the R square value is also very less and hence, the overall utility
of the model is very less.
(c) Inserting a dummy variable
Code = Morning and early afternoon =0
Code = Afternoon = 1
(d) It is apparent that multiple regression model (boxes, weight and codes) are present is
more appropriate because the p value for all the slope coefficient is zero which means
they are significant. Further, the R square value is very high that shows that that the
predictive power of this model is higher than model of part a. Hence, Model b would be
considered as better.
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(e) Slope coefficient (time of day) is significant or not.
Hypothesis testing
Ho : Slope is insignificant.
HaSlope is significant.
Null hypothesis will be rejected when the p value of the slope coefficient is lower than
significance level (0.05). Here, the p value for slope coefficient (time of day) is zero which
means it is lower than significance level. Thus, sufficient evidence is present to reject null
hypothesis. Hence, Slope coefficient (time of day) is significant in relation to unload the
boxes.
(f) Time needed to unload the trucks =?
Total boxes = 100
Weight = 5000 kg or 50 (hundred kg)
Times of day = 1 (morning), 2(early afternoon), 3(late afternoon)
Time = −41.422 + ሺ0.644 ∗ Boxesሻ+ ሺ0.349 ∗ Weightሻ+ (4.543 ∗ Codes)
Case 1: Time need to unload the truck hen the time of day is morning (put code =1)
Time = −41.422 + ሺ0.644 ∗ 100ሻ+ ሺ0.349 ∗ 50ሻ+ ሺ4.543 ∗ 1ሻ= 45.01 minutes
Case 2: Time required to unload truck in the early afternoon (code =2)
Time = −41.422 + ሺ0.644 ∗ 100ሻ+ ሺ0.349 ∗ 50ሻ+ ሺ4.543 ∗ 2ሻ= 49.55 minutes
Case 3: Time required to unload truck in the late afternoon (code =3)
Time = −41.422 + ሺ0.644 ∗ 100ሻ+ ሺ0.349 ∗ 50ሻ+ ሺ4.543 ∗ 3ሻ= 54.09 minutes
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