Descriptive Statistics And Hypothesis Testing

Descriptive statistics of demographic variable of the sample

Descriptive statistics of demographic variable of the sample:

For age:

Table 1: Frequency table for Age

Categories	Frequency Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper
Others	1
35 or less	20
36-45	45
46-55	51
56-65	18
65+	10
Missing data	10
Total	145

Fig 1: Pie chart for Age.

The age range starts from 0 years to more than 65 years. It can be seen from the frequency table and the pie chart that people with ages in the range of 46 – 55 has the maximum frequency that is of 51 the second highest frequency is of people in the age range 36 – 45. The lowest frequency is of the group with age 65+ (Sharpe  2015). Rest of the groups of ages falls in middle. There are few missing in the age data set.

For Sex:

Table 2: Frequency table for Sex

Categories	Frequency
Female	57
Male	88
Missing value	10
Total	155

Fig 3: Pie chart for Sex.

The sex range is divided into male and female. It can be seen from the frequency table and the pie chart that people that female population has the maximum frequency that is of 88  and the frequency of male population is a little less than this that is of 57 (Shi  2018). There are few missing in the age data set.

For Education:

Table 3: Frequency table for education.

Category	Frequency
Less than HS	2
High school diploma	13
Some college	34
College grads	62
Graduate degree	33
Missing system	11
Total	155

The education range starts from less than high school to graduate degree. It can be seen from the frequency table and the pie chart that people who are college grads has the maximum frequency that is of 62 and  the second highest frequency is of people who are some college grads and the frequency is 34. The lowest frequency is of the group of people whose highest degree is less than HS.. Rest of the groups falls in middle. There are few missing in the age data set.

Figure 4: Pie chart for Education.

For Income:-

Table 5: Frequency table for Income.

Category	Frequency
35K – 50K	10
50K – 65K	8
>65K	123
Missing system	14
Total	155

Figure 5: Pie chart for income.

The income range starts from 35K to more than 65K. It can be seen from the frequency table and the pie chart that people with income in the range of more than 65K has maximum frequency and has the maximum percentage in the whole population (Kenny  2014). The lowest frequency fall in the group 35K – 50K and with 5% people from the population. Another group with income rage of 50K – 65K  and with 7%people in the whole population falls in the middle.  There are few missing in the age data set.

Proposition and testing of hypothesis

For Cars:

Table 6: Frequency table for Cars.

Category	Frequency
American	58
European	38
Japanese	59
Total	155

Figure: Pie chart for Car.

The car range has three divisions and they are American, European and Japanese (Manrai  2016). It can be seen from the frequency table and the pie chart that people with Japanese car has the highest frequency that is of 59 and consist of 38%of the population. People with American cars have the second highest frequency that is of 58 and consist of 37% of the population. People with European cars have the lowest frequency that is of 38 and consist of 25% of the whole frequency.

Proposition and testing of hypothesis:-

Cross-tabs for categorical vs categorical data:-

1. Sex and Education:

A crosstab has been formed with the two categorical variable sex and education. The two level of sex that is male and female are being kept in row division (Farg  and Khalil  2015). The five levels of education that is less than HS, high school diploma, some college, college grads and graduate degree are being divided according to the rows. Cross-tab for the divisions is shown below:

Table 7: Cross tab for sex and education.

Independent Samples Test
	Levene’s Test for Equality of Variances	t-test for Equality of Means
F	Sig.	t	df	Sig. (2-tailed)	Mean Difference	Std. Error Difference	95% Confidence Interval of the Difference
Lower	Upper
educ	Equal variances assumed	6.571	.011	2.405	142	.017	.38474	.15998	.06849	.70099
Equal variances not assumed			2.321	103.532	.022	.38474	.16579	.05595	.71353

It can be clearly seen from the table that the female population has a single person with less than Hs degree, 8 people with high school diploma, 18 people with degree of some college, 158 people are college grads and 11 people has the graduate degree.

Similarly, it can be clearly seen from the table that the male population has a single person with less than Hs degree, 5 people with high school diploma, 16 people with degree of some college, 44 people are college grads and 22 people has the graduate degree (Pandis  2016).

The data can be further tested for a relation between the two variable with chi-square test.

Test details are:

Test hypothesis:-

H₀ : There is no association between the two variables.

Vs.

H₁ : The is association between the two variables.

Test statistic:  

Table for the test is given below:

Table 8: Chi square test table for Sex and education.

Chi-Square Tests
	Value	df	Asymp. Sig. (2-sided)
Pearson Chi-Square	8.698a	4	.069
Likelihood Ratio	8.631	4	.071
Linear-by-Linear Association	5.597	1	.018
N of Valid Cases	144
a. 2 cells (20.0%) have expected count less than 5. The minimum expected count is .78.

It can be seen from the table that calculated chi square is 8.698 and significant chi square is 0.069 which is less than tabulated one. Therefore, the null hypothesis can be rejected here and it can be said that the two variables are not independent.

1. Car and sense of accomplishment:

A crosstab has been formed with the two categorical variables car and sense of accomplishment (Koletsi and Pandis 2016). The three categories of car that is American, European and Japanese are being kept in row division. The 7 levels of sense of accomplishment that is 1,2,3,4,5,6 and 7 are being divided according to the rows. Cross-tab for the divisions is shown below:

Table 9: Cross tab for cars and sense of accomplishment

car * sense of accomplishment Crosstabulation
Count
	sense of accomplishment	Total
1.00	2.00	3.00	4.00	5.00	6.00	7.00
car	American	12	20	11	12	0	1	2	58
European	9	14	9	1	2	1	1	37
Japanese	12	18	18	0	2	5	3	58
Total	33	52	38	13	4	7	6	153

It can be clearly seen from the table that among people with American cars, 12 people has satisfaction degree 1, 20 people has satisfaction degree 2, 11 people has satisfaction degree, 12 people has satisfaction degree 4, 0 people has satisfaction degree 5, 1 person has satisfaction degree 6 and 2 people has satisfaction degree 7.

Again, it can be clearly seen from the table that the among people with European cars, 9 people has satisfaction degree 1, 14 people has satisfaction degree 2, 9 people has satisfaction degree, 1 people has satisfaction degree 4, 2 people has satisfaction degree 5, 1 person has satisfaction degree 6 and 1 person has satisfaction degree 7.

Similarly, it can be clearly seen from the table that the among people with Japanese car, 12 people has satisfaction degree 1, 18 people has satisfaction degree 2, 18 people has satisfaction degree, 0 people has satisfaction degree 4, 2 people has satisfaction degree 5, 5 person has satisfaction degree 6 and 3 people has satisfaction degree 7.

The data can be further tested for a relation between the two variable with chi-square test.

Test details are:

Test hypothesis:-

H₀ : There is no association between the two variables.

Vs.

H₁ : The is association between the two variables.

Test statistic:  

Table for the test is given below:

Table 9: Chi square test table for Car and level of satisfaction.

Chi-Square Tests
	Value	df	Asymp. Sig. (2-sided)
Pearson Chi-Square	25.297a	12	.013
Likelihood Ratio	29.006	12	.004
Linear-by-Linear Association	.370	1	.543
N of Valid Cases	153
a. 12 cells (57.1%) have expected count less than 5. The minimum expected count is .97.

It can be seen from the table that calculated chi square is 25.297 and significant chi square is 0.13 which is less than tabulated one. Therefore, the null hypothesis can be rejected here and it can be said that the two variables are not independent.

T-test for categorical vs scale variable:-

1. For age and sex:

Hypothesis are:

H₀:  mean of age is equal to mean of sex..

Vs.

H₁ : mean of age is not equal to mean of sex..

Test statistic:-  (x? – µ)/ σ/√n.

Table of calculation here is given below:-

Table 10:- T-test.

Independent Samples Test
	Levene’s Test for Equality of Variances	t-test for Equality of Means
F	Sig.	t	df	Sig. (2-tailed)	Mean Difference	Std. Error Difference	95% Confidence Interval of the Difference
Lower	Upper
age	Equal variances assumed	2.568	.111	.361	143	.719	.06778	.18787	-.30358	.43915
Equal variances not assumed			.349	106.487	.728	.06778	.19415	-.31712	.45269

It can be seen from the table that calculated t is 0.361 with the assumption of equal variance and 0.349 without the assumption of equal variance. Again, significant t is 0.719 with equal variance and 0.728 with unequal variance. Therefore, significant t is less than tabulated t in both the cases and therefore the null hypothesis can be accepted in both the cases.

1. For Sex and education:

Hypothesis are:

H₀:  mean of age is equal to mean of education.

Vs.

H₁ : mean of age is not equal to mean of education.

Test statistic:-  (x? – µ)/ σ/√n.

Table of calculation here is given below:-

Table 10:- T-test.

Independent Samples Test
	Levene’s Test for Equality of Variances	t-test for Equality of Means
F	Sig.	t	df	Sig. (2-tailed)	Mean Difference	Std. Error Difference	95% Confidence Interval of the Difference
Lower	Upper
educ	Equal variances assumed	6.571	.011	2.405	142	.017	.38474	.15998	.06849	.70099
Equal variances not assumed			2.321	103.532	.022	.38474	.16579	.05595	.71353

It can be seen from the table that calculated t is 2.405 with the assumption of equal variance and 2.321 without the assumption of equal variance. Again, significant t is 0.719 with equal variance and 0.022 with unequal variance (Helwani  2015). Therefore, significant t is less than tabulated t in both the cases and therefore the null hypothesis can be accepted in both the cases.

ANOVA for categorical vs scale variable:

1. Anova test for income and car can be written as:

Hypothesis are:

H₀:  mean of income is equal to mean of car.

Vs.

H₁ : mean of income is not equal to mean of car.

Test statistic:-  Mean square between group/mean square of error.

The analysis table is:

Table 11: ANOVA of car and income.

ANOVA
income
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	(Combined)	1.182	1	1.182	3.984	.048
Linear Term	Unweighted	1.182	1	1.182	3.984	.048
Weighted	1.182	1	1.182	3.984	.048
Within Groups	41.257	139	.297
Total	42.440	140

It can be seen here the test statistic value is 3.984 and the significant value is 0.48 which is less than statistic value (Burger et al.  2015). Therefore, the null hypothesis can be rejected here and it can be predicted that the means are not equal.

1. Anova test for income and age can be written as:

Hypothesis are:

H₀:  mean of income is equal to mean of age.

Vs.

H₁ : mean of income is not equal to mean of age.

Test statistic: – Mean Square between group/mean square errors.

The analysis table is:

Table 11: ANOVA of car and income.

ANOVA
income
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	(Combined)	1.581	2	.791	2.671	.073
Linear Term	Unweighted	.236	1	.236	.797	.374
Weighted	.236	1	.236	.797	.374
Deviation	1.346	1	1.346	4.545	.035
Within Groups	40.858	138	.296
Total	42.440	140

It can be seen here the test statistic value is 2.671 and the significant value is 0.073 which is less than statistic value (Yang et al.  2014). Therefore, the null hypothesis can be rejected here and it can be predicted that the means are not equal.

Correlation for scale vs. scale:-

For sex and issue 1:-

Hypothesis are:

H₀:  They are not correlated.

Vs.

H₁ : They are correlated.

Test statistic: –  r*√(n-2)/√(1-r^2).

The table for the test is given below:-

Table 12: Table for correlation test

Correlations
	sex	major part of my fun
sex	Pearson Correlation	1	-.216**
Sig. (2-tailed)		.009
Sum of Squares and Cross-products	34.593	-26.278
Covariance	.240	-.184
N	145	144
major part of my fun	Pearson Correlation	-.216**	1
Sig. (2-tailed)	.009
Sum of Squares and Cross-products	-26.278	456.260
Covariance	-.184	2.982
N	144	154
**. Correlation is significant at the 0.01 level (2-tailed).

It can be seen here that value of pearsonian statistic is  -0.027 and the significant value is 0.784 which is greater than the value of statistic. Therefore, the null hypothesis can be accepted here and it can be said that there is no correlation between issue 1 and age.

Bivariate regression for scale verses scale.

1. For being good to myself and major part of my fun:-

Hypothesis are:

H₀:  They are not related.

Vs.

H₁ : They are related.  

Table for test results is:-

Table 13: Regression table.

ANOVAa
Model	Sum of Squares	df	Mean Square	F	Sig.
1	Regression	147.535	1	147.535	72.639	.000b
Residual	308.724	152	2.031
Total	456.260	153
a. Dependent Variable: major part of my fun
b. Predictors: (Constant), being good to myself

It can be shown from the table that the value of f statistic is 72.639 and its significant value is 0.766 Which is smaller than the f statistic value (Näsman  2015). Therefore, there exist a relation between being good to myself and major part of my fun.s

The regression line can be calculated here. The table for co-efficient is given below:

Table 14: Table for coefficients. 

ANOVAa
Model	Sum of Squares	df	Mean Square	F	Sig.
1	Regression	147.535	1	147.535	72.639	.000b
Residual	308.724	152	2.031
Total	456.260	153
a. Dependent Variable: major part of my fun
b. Predictors: (Constant), being good to myself

From the table of coeffic8ient it can be seen that the tabulated t value of being good to myself is 8.523 and the significant t value is 1.96 which is smaller than the significant t. therefore, the null hypothesis can be rejected and the factor can be accepted here. Therefore, the regression line is:

 major part of my fun = 2.147 + 0.509 * being good to myself.

1. For age and being good to myself:

Hypothesis are:

H₀:  They are not related.

Vs.

H₁ : They are related.

The required regression table is:

Table 14: Regression table. 

ANOVAa
Model	Sum of Squares	df	Mean Square	F	Sig.
1	Regression	.079	1	.079	.065	.799b
Residual	174.679	143	1.222
Total	174.759	144
a. Dependent Variable: age
b. Predictors: (Constant), being good to myself

It can be seen from the table that the tabulated F value is 0.065 and the critical F value is 0.762 which s greater than the calculated value. Therefore, the null hypothesis can be rejected here and it can be said that the two variables are related.

The regression line can be calculated here. The coefficient table is attached below:

Table 15: Coefficient table.

Coefficientsa
Model	Unstandardized Coefficients	Standardized Coefficients	t	Sig.
B	Std. Error	Beta
1	(Constant)	3.615	.183		19.767	.000
being good to myself	.012	.048	.021	.255	.799
a. Dependent Variable: age

It can be seen from the table that the t statistic value is 19.767 and the t significant value is 1.96, which is smaller than t significant value. Therefore, the coefficient value of the factor can be  accepted here and the regression line can be calculated as

Sex = 3.615 + 0.012*being good to myself.

Interpretation of the findings:

It can be interpreted here that sex and education are interrelated that is the level of education is related with sex. Sense of accomplishment and cars are related.. Again, the means between age and sex are the same and the means between sex and education are not same. There may not be an equal division of the two sex regarding age groups and education is also not divided equally among the two sex group.  The mean income and the mean age are also not the same. Mean income and mean purchase of the category of cars are not the same. There is no correlation between being good to myself and age and there exists a relation between good to myself and main reason of my fun. Age and being good to myself are also related to each other.

Conclusion:

There are numbers relations that can be explained with the study. Member of a particular sex have an easy assess for education and it is not that easy an assess for members of the opposite sex. People with higher education levels can be proved to be more intellectual employees. People with a particular type of car are more accomplished than the other and can increase the sale of that car type. Sex is not equally divided in the different age groups and all of the age divisions does not earn equally. Therefore, this can effect demand of few product which are sex centric. Earning cause a huge impact on purchase and demand. Age is not related to being good to one’s own self but being good to one’s own self is related to having fun. The fun factor can effect purchase market as fun can be materialistic.

It can be recommended here that since the sex division is not equal, market should focus more on demand type of the dominant sex. Company should also focus on hiring intellectual employee that is educated employee which is also dependent on sex.

References

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Kenny, D.A., 2014. Measuring model fit.

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