Hypothesis - Practical Examples

Caltex has 550 retails stations in Pakistan and they are having a facility High Speed Diesel (HSD), Motor Gasoline (Mogas), Compressed Natural Gas (CNG) and Lubricants. We have conducted a research on the selling volumes per month in the Pakistan. We found out that the Volume Population Mean of 550 Retail Stations as 143 KL per month and the population Standard deviation as 118.6 KL per month. Now we have taken the random sample of 60 Sites from different cites like Karachi, Islamabad and Lahore to mitigate the error in the data. The sample mean of the 60 sites is found as 199 KL per month and the sample standard deviation of the volume of the retail stations is…….

Case I

 We are going to test the hypothesis that whether the population mean volumes of the retail outlets of Pakistan is greater than 143KL per month, according to our given random sample.  

Per Month Volumes of 60 random Retail Outlets in KL from Karachi, Islamabad and Lahore
120	101	87	358	179	146
547	73	329	308	83	331
359	68	133	76	378	161
103	209	112	327	185	125
218	299	78	133	128	314
334	252	179	96	211	515
110	85	46	404	173	177
371	172	91	264	113	151
236	186	62	573	51	69
113	104	366	66	113	185

Testing of Hypothesis

Testing of Hypothesis that at 5% significance level, the population mean volumes of the retail stations in Pakistan is greater than 143 KL per month.

Step # 1

µ = 143KL per month

µ > 143KL per month

Step # 2

Level of Significance = α=0.05

Step# 3

Critical Points: Z_tab

Z_{tab =} Z_0.05= 1.645

Step# 4

                                                                                      __

Z_cal= (X - µ) / (σ/ )

 =   (199-143) / (118.6)/

                                       =     56 / 15.311

                                       =     3.65

Step# 5

If  │Z_cal│ > │Z_tab│, then reject Ho.

=  │3.65│ > │1.645│

In this scenario Z_cal > Z_tab then reject Ho.

Conclusion:

At 5% of significances level we have enough evidence to conclude that the population mean volumes of the retail outlets of Pakistan is greater than 143KL per month.

Case II

We have used the Chi-square test to test for independence between rows and columns of a contingency table. Here is a real life example.

In Pakistan, hydrocarbon is now everyone needs either it is directly or indirectly. So, we have chosen the study that does the area effect the fuel consumption of the fuel station. We have separated the fuel stations by province wise and further categorized them as fuel sales wise. The results are summarized in the following 4X4 contingency table:

Test of Independence

Test of Independence		Petrol Pumps of Caltex in Pakistan by State
		Sind	Punjab	Baluchistan	NPKP & Northern Areas	Total
Volumes in KL	Less Than 100 KL	80	130	4	30	244
	From 100 KL to 200 KL	45	89	5	36	175
	From 201 to 300 KL	20	27	8	16	71
	From 301 and Above	15	24	10	11	60
	Total	160	270	27	93	550

Step# 1

                        Ho:    Area doesn’t effect on the sale (Volumes) of the petrol pumps

                        H1:    Area does matter on the sale (Volumes) of the petrol pumps.

Step# 2

         E_ij = Row total x Column Total / Grand Total

Observed Frequency (O)	Expected Frequency (E)	Chi Square   = (O - E)2/E
80	70.98	1.146
148	119.78	6.648
4	11.98	5.314
30	41.26	3.072
45	50.91	0.686
93	85.91	0.585
2	8.59	5.057
36	29.59	1.388
20	20.65	0.021
27	34.85	1.770
1	3.49	1.772
16	12.01	1.329
15	17.45	0.345
24	29.45	1.010
1	2.95	1.285
8	10.15	0.454
Total Chi Square Calculated		31.881

Step# 3

Assumptions:

·                All the expected frequencies must be 1 or greater.

·                At most 20% of the expected frequencies must be less than 5.

 Both the conditions are fulfilled in the selected scenario.

Step# 4

Level of Significance = α=0.05

Step# 5

Critical Points: χ²_tab at 0.05 level of significance

Degree of Freedom = (R-1) x (C-1)

 =  (4-1) x (4-1)

                                               =   9

    χ²_tab at 0.05 level of significance with 9 degree of Freedom is 16.9

Step# 6

χ²_cal = 31.881

Step # 7

If χ²_{cal >} χ²_tab, the reject Ho.

In this scenario χ²_{cal >} χ²_tab

                                                31.881    >                 16.9

So, reject Ho.

Conclusion:

That means we have sufficient evidence to conclude that at 5% significance level Area does matter on the sale (Volumes) of the petrol pumps.