The observations were as follows:
1. Relation of time devoted to self-study with the monthly pocket money:
An inverse relation is evident, when plotting monthly pocket money against the time devoted to self-study (hrs per week). It was observed that on an average, college students receiving pocket money around Rs.500 per month utilize about 30-50 hrs per week for self-study. On the other hand, as the pocket money rises, study time falls to as low as 5hrs per week. This analysis gives us an insight into student psychology. It is certain that students get diverted with too much money in hand, and education takes a backseat.
2. Relation between the number of movies seen in a month with the pocket money received:
A clearly incremental graph is obtained when the number of movies seen (per month) is considered alongside the monthly pocket money. With monthly pocket money of Rs.500 an average student watches about 2 movies per month, while one receiving Rs.15000 can afford to watch more than 10. From this we can infer that movies being a common expenditure of college students, the more money is available, the more movies are seen per month.
3. Variation of average pocket money on the basis of gender:
52 females and 51 males provided the 103 data entries. This uniformity provides us the opportunity to obtain a value for their respective average monthly pocket money. It is noticeable that an average female receives a higher pocket money than an average male. While the average pocket money of males is close to Rs.3000, that of an average female is more than Rs.3500.
4. Variation in time devoted to self-study on the basis of gender:
It is a realised trend that females usually devote more time to self-study than males. However, the results of our analysis do not conform to it. As per our analysis males append about 19hrs per week on self-study, while females approximately 13hrs. As per our analysis, it was observed that females receive a higher pocket money than males and also that with an increase in pocket money the time devoted to self-study decreases. This could be a possible cause for the obtained result and deviation from the trend.
5. Relation of Monthly Family Income with Monthly Pocket Money:
Distribution is always with respect to a whole. Similarly, the apparent fact of pocket money given to children being dependent on the income of the family is verified by the graph below. It is clear from the graph that pocket money follows a direct relation with the family income.
INFERENCIAL ANALYSIS
Inferential statistics basically relates the sample characteristics to population characteristics. We take random samples from the populations because we cannot take the entire populations for the purpose of our study. Samples are simply the means to an end and not an end in themselves. Statistical inference is concerned with making inferences about the population parameters. Typical population parameters are the mean, standard deviation, the area under the curve between two values of a variable etc.
Inferences about Regression as a whole: Overall Significance
Using F-test
The F test is used to determine whether there is a significant relationship between the dependent variable and the set of all the independent variables. It helps us check whether the value of R-square really indicated that the independent variables explains Y or has happened just by chance.
“Is the regression as a whole significant?”
We see how all the Xi’s taken together significantly explain the variability explained in Y. Our hypotheses is
Ho: B1 = B2 = B3 ….. = Bk
Null hypotheses: Y does not depend on Xi’s.
Hi: At least one Bi≠0
Alternate Hypotheses: Y depend upon at least one Xi’s
In discussing the Variation in Y we look at three different terms, we denote these by:
Three different Sums of Squares,
These are related by Equation
“Decomposing the total Variation in Y”
SST = SSR + SSE
Which says that the total variation in Y can be broken down into two parts :
Explained Part & Unexplained part.
Each of these squares has associated (n-1) degrees of freedom (n observation 1) degrees of freedom because sample mean is fixed. SSR has k Degrees of Freedom because there are ‘k’ independent variables being used to explain Y.
Finally, SSE has (n-K-1) degrees of freedom because we used our ‘n’ observation to estimate
(K+1) constants, a, b1, b2…………..bk.
If the null hypothesis is true the ratio below is equal to.
From the summary output: We see that SSR = 727000000 (with K=7 degrees of freedom)
SSE = 378000000(with n-k-1=102-7-1= 95 degrees of freedom)
So, F = 727000000/7
378000000/95
= 26.14
the entries in the “MS” column are just the sums of squares divided by degrees of freedom for 7 numerator degrees of freedom & 95-denominator degree of freedom. The table tells us that 2.12 is the upper limit of acceptance region. For a significance level of
α = 0.05.
Our calculated F value of 26.14 is far above 2.13 so we see that regression as a whole is highly significant. We can reach the same conclusion by noting that the output tells us that “significance F”= 1.31e-19
Inferences about an Individual Factor: Individual Significance
The z-test
The T Test is used to determine whether each of the individual independent variables is significant. A separate t test is conducted for each of the independent variables. The purpose of the t-test is to see whether we can use the sample data to conclude that
The regression plane is derived from a sample and not from the entire population. As a result, we cannot expect the true regression equation:
Y = A+ B1X1+B2X2+………..+BkXk
(The one for the entire population)
To be exactly the same, as the equation estimated from the sample observations,
Y = a + b1X1 + b2X2 +.…….+ bkXk
Even so we can use the value of b1, one of the factors, we calculate from the sample, to test hypotheses value of B1, one of the factors of the regression plane for the entire population.
We test the following Hypothesis about B1.
H0 : B1 = 0
Null Hypotheses
H1 : B1 ≠ 0
Alternate Hypotheses
If H0 is rejected, we will conclude that B1 ≠ 0 and that the two variables have a statistically significant relationship. However, if H0 cannot be rejected [We reject H0 if (t < -tα/2) or if (t > tα/2)], we would not have sufficient evidence to conclude that there is a relation between the variables.
Age:
Here,
test statistic, t = 2.891083
t0.025 = 1.50
With 2.891083>1.50 (t0.025), we reject H0 and conclude at the 0.05 significance that B1 is not equal to zero.
The statistical evidence shows that there is a significant relationship between age and monthly expenses.
Number of working parents:
Here,
test statistic, t = 2.155667
t0.025 = 1.50
With 2.155667>1.50 (t0.025), we reject H0 and conclude at the 0.05 significance that B1 is not equal to zero.
The statistical evidence shows that there is significant relationship between Number of working parents with monthly expense of students.
Monthly income:
Here,
test statistic, t = 2.613987
t0.025 = 1.50
With 2.613987<>.50 (t0.025), we reject H0 and conclude at the 0.05 significance that B1 is not equal to zero.
The statistical evidence shows that there is significant relationship between family monthly income and monthly expenses of students,
Number of siblings:
Here,
test statistic, t = -0.902742
t0.025 = 1.50
With –0.902742<1.50 (t0.025), we accept H0 and conclude at the 0.05 significance that B1 is equal to zero.
The statistical evidence shows that there is no significant relationship between number of siblings and monthly expenses of students,
Number of movies seen in a month:
Here,
test statistic, t = 3.142832
t0.025 = 1.50
With 3.142832>1.50 (t0.025), we reject H0 and conclude at the 0.05 significance that B1 is not equal to zero.
The statistical evidence shows that there is significant relationship between Number of movies seen in a month and monthly expenses of students.
Visits to restaurants per month:
Here,
test statistic, t = 5.421227
t0.025 = 3.96
With 5.421227>1.50 (t0.025), we reject H0 and conclude at the 0.05 significance that B1 is not equal to zero.
The statistical evidence shows that there is significant relationship between Number of restaurant visits per month and monthly expenses of students.
Time devoted to self study:
Here,
test statistics, t = -1.747169
t0.025 = 3.96
With –1.747169<1.50 (t0.025), we accept H0 and conclude at the 0.05 significance that B1 is equal to zero.
The statistical evidence shows that there is no significant relationship between time devoted to self study and monthly expenses of students.
Also, we use the P-value to test whether the factors are significant explanatory variables or not. The entries in the column are probability values for the two-tailed test of the hypotheses.
Ho: B3 = 0
H1: B3 ≠ 0
We only compare the probability value with α the significant level of the test, to determine where QF is a significant variable of Y.
Testing the significance of an explanatory variable is always a two-tailed test. The independent variable Xi is a significant explanatory variable if “bi” is significantly different from zero, that is, if to is a large positive or a large negative number.
Testing Hypothesis for independence of two categories though χ2
CHI square test
CHI square is used as goodness of fit test, it is most of the used as a test of independence to determine if the paused observations obtained on two or more nominal variables are independent of each other or not. It is some times necessary to deal with the idea of two variables being related in the sense that the value of one variable depends upon the value of other corresponding variable.
Then the null hypothesis for independence can be tested by: -
χ2 = ∑ (fo – fe)2
fe
Where:
Fo = Observed frequency
Fe = Expected frequency = row total*column total/100
∑ = summation overall cells
The degrees of freedom in a contingency table are given below;
df = (r-1) (k-1)
Where:
R = number of rows
K = number of columns
Relationship Between Family Income and Fathers Education.
H0 = 0: Null Hypothesis: no relationship between income and education.
H1 = Alternate Hypothesis: relationship between income and education.
Solving the problem we get the value of χ2 as 6.8465
Looking at the value of χ2 from the table for 95% confidence (or α = 0.5) & at (6-1)(2-1) = 5 degrees of freedom, we get:
χ2 = 11.0705
Since our calculated value of χ2 is less than the critical value of χ2 i.e. 11.0705; we accept the null hypothesis and reject Alternate hypothesis, concluding that there is no relationship between family income and number of working members in the family.
Relationship between amount of time devoted to self study and number of movies seen in a month
H0 = 0: Null Hypothesis: no relationship between time spent on self study and number of movies seen in a month.
H1 = Alternate Hypothesis: relationship between time spent on self study and number of movies seen in a month
Solving the problem we get the value of χ2 as 5.1829
Looking at the value of χ2 from the table for 95% confidence (or α = 0.5) & at (5-1)(4-1) =12 degrees of freedom, we get:
χ2 = 21.0261
Since our calculated value of χ2 is less than the critical value of χ2 i.e. 21.0261; we accept the null hypothesis and reject Alternate hypothesis, concluding that there is no relationship between number of movies seen in a month and time spent on self study.
EXCEL FUNCTIONS USED
Alongside the essential functions of regression, t-test, f-test, chi square test, mean, certain other functions were also used to make interpretations and calculations easier. An example of this is the sort function.
Use of ‘Sort’ Function:
The Y factor, namely the monthly pocket money was arranged in ascending order. This Excel function provided the convenience of analysing whether the sample space had been appropriately selected, so as to represent the population. The data collected was significant as it included a broad range that varied between Rs. 500 to Rs. 15000 per month. Also was identifiable a gap in the incremental trend of the data collected. The data does not include any entries with pocket money ranging from Rs. 5500 to Rs. 8000. A method to rectify this is by entering all useful data simultaneously as it is obtained. However, in this research the impact of this drawback is reduced as the number of questionnaires considered was many. i.e. 103
This function was also used to segregate the data on a gender basis. Dummy variables were used to depict this variable. Females were represented by 1, and the males with a 0 as for binary coding. These values were then sorted to obtain a data segregated on the basis of gender. This made further analysis on this basis easier.
CONCLUSION
From a sample space of around 100 students we made inferences about the various factors influencing the pocket expenses of students. Subsequent to the analysis, it was notices that certain factors have a direct relation with monthly pocket money while others demonstrate an inverse relation.
Variables with direct relation:
As this questionnaire was targeted at the college students the age group selected was 17 – 24. It was observed that in this age group your pocket expenses increase with age as a rise was observed in the expenditure on education and entertainment.
It was observed that both parents working, leading to greater family income had a direct impact on the pocket money of students. Also these students were observed to be more outgoing and spent major part of their pocket expense on entertainment.
It is however obvious that greater the pocket money greater would be the money spent on entertainment. As in the questionnaire we divided entertainment in various sub heads such as movies, eating out, money spent on gym and health club etc, we observed the major portion of money spent was entertainment.
Variables with Indirect Relation:
From the observed data we concluded that more the number of siblings lesser is the pocket many i.e the number of siblings and the monthly pocket money of students has an inverse relation. It is thus seen the more the number of siblings, the money gets distributed hence lesser money for each student.
- Time devoted to self-study:
Our analysis showed that students who spend more time on self study spend less on entertainment and hence have quite reasonable monthly expenses. Thus an inverse relation is seen between both variables.
APPENDIX
The Questionnaire:
FACTORS INFLUENCING POCKET EXPENSES OF COLLEGE STUDENTS
All the information provided by you will be strictly used for academic purposes only and will be kept confidential.
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1. Who is/are the working parent(s) in your family?
2. What is the highest level of education of your parents?
3. What is the main source of your family income?
4. What is your family’s average monthly income? (From all sources)
5. How many siblings do you have?
6. What is your average monthly pocket money?
7. Do you own a personal vehicle?
8. How many movies do you watch in a month? _____ (per month)
9. How many times do you visit to a restaurant/eating joints in a month? _____ (visits per month)
10. Do you visit a fitness centre (gym/health club)?
If yes, how many days in a week _____ (days per week)
11. How much time do you devote to self study? _____ (hours per week)
Signature: _____________
Date: _________________
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