Acknowledgements

First of all, I would like to thank God for giving me the life and good health needed to

carry out this dissertation.

I would also like to thank my teacher Mrs. Scott for giving me guidance in putting

together this SBA.

Moreover I am very grateful to the staff of the JnMarie & Sons Ltd. for providing me

with the necessary information. In particular, I would like to thank the co-owner, Mr. Everistus

Jn-Marie, for not only providing me with information but answering my questions and queries.

Finally, I would like to thank my parents, for providing me with the resources needed to

complete this SBA.

1 | P a g e

Introduction

JnMarie & Sons is a private limited company that was formed in 1977 by Raphael Jn-Marie. The

gas station has two branches in Castries, one on Manuel Street and the other on the John

Compton Highway.

One of the products that JnMarie & Sons sells is Liquefied Petroleum Gas (LPG) which is more

commonly known as cooking gas. The business orders LPG in large 6000 lb cylinders called

Iso-Tanks and then transfers it to smaller 20 lb cylinders for sale.

One Iso-Tank costs $21,710 whilst the company sells each 20 lb cylinder at $39.60.

The Iso-Tank is supposed to fill a total of 1,300 smaller 20 lb cylinders. However, the gas station

often gets more or less than this proposed amount of gas.

Therefore, the question is, on average, does an Iso-Tank fill 1,300 smaller 20 lb cylinders? This

project is aimed at answering this question.

2 | P a g e

Purpose and Aims

The main purpose of this project is to find out whether if, on average, a 6000 lb cylinder of LPG

ordered by JnMarie & Sons fills the correct amount of 20 lb cylinders.

The project has the following aims:

To use a hypothesis test to determine whether or not, on average, an Iso-Tank of LPG

fills 1,300 smaller 20 lb cylinders.

To estimate a 95% confidence interval for the mean amount of 20lb cylinders that are

filled from an Iso-Tank.

To identify outliers in the iso-tank outputs and investigate the reasons for the existence

of these outliers.

3 | P a g e

Variables

1. Iso-Tank Output (No. of 20lb Cylinders)

The number of 20 lb cylinders filled from each iso-tank varies from tank to tank.

Factors Affecting Iso-Tank Output

Below are several factors that can affect the number of 20 lb cylinders filled from a 6000

lb tank. These are:

i. Leakages

Leakages of the 6000 lb tanks can occur during transit or whilst the gas is being

transferred to the 20 lb cylinders. Significant leakages would result in less 20 lb cylinders

being filled.

ii. Theft & Sabotage

Employees or other personnel can steal gas from 6000 lb tank for their personal use or

to sabotage the company. This would reduce the number of 20 lb cylinders filled.

iii. Amount of LPG Left in Returned Tanks

If cylinders received from customers for replenishment has some unused product then it

would take less gas from the Iso-Tank to fill the 20 lb cylinder. Hence, more gas will be

available to fill other tanks, which may cause the output being of 1300 cylinders.

iv. Manufacturing Procedures

The process used to fill the 6000 lb tanks by the manufacturer will affect the number of

filled 20 lb cylinders received by Jn Marie. If the manufacturing procedures are good and

consistent, then the correct amount of gas should be obtained in most 6000 lb cylinders.

However, if it is faulty, greater or lesser amounts may be placed into the 6000 lb

cylinders.

4 | P a g e

Unfortunately, it was not possible to control any of the above mentioned factors.

NB: Errors made by employees in counting the number of cylinders can also cause the iso-

tank output to vary.

5 | P a g e

Data Collection

1. Method of Data Collection

The method used was documentary research. Co-owner of the firm, Mr. Everistus Jn-

Marie had the most recent figures written on a board in his office. Hence, on Monday

10th March 2014, I went to his office and took down these figures by hand.

It would have been much too impractical to collect the data directly. This is because it

would take a significant amount of time to count thousands of 20 lb cylinders.

Moreover, the gas station does not order several 6000 lb tanks at once, but once every

week. Therefore, in order to obtain a sufficiently large sample of 30 measurements it

would take 30 weeks.

However, the process used by employees to count the number of filled tanks is

explained below:

At the filling area, empties are counted before filling begins.

They are batched according to their tare weight (weight of an empty tank).

The scale is set accordingly and the tanks are filled, two tanks at a time.

E.g. If a tank has a tare weight of 25 lbs and 20lbs of product needs to be placed

into it then the scale is set at 45lbs during filling. The filling of the tank will

automatically cease at the weight of 45lbs.

2. Sampling

Quota sampling was used mainly due to the time constraint. The sample used was the

set of figures obtained from the board in Mr. Jn-Marie’s office. These figures were the

67 most recent counts of the number of 20 lb cylinders filled from a 6000 lb tank.

6 | P a g e

Below: Refined Stem & Leaf Diagram Illustrating Iso-Tank Outputs

Stem Leaf Key: 123/6 means 1236

122 9 (1)

123 6 (1)

124 0 (1)

125 7 7 7 8 (4)

126 1 7 (2)

127 1 3 4 5 (4)

128 0 1 1 3 3 4 5 9 (8)

129 0 0 0 0 1 2 3 5 8 8 8 9 (12)

130 0 1 2 3 4 4 5 6 6 9 (10)

131 0 0 2 2 4 6 6 8 9 9 (10)

132 0 1 1 2 2 6 7 8 8 (9)

133 0 5 (2)

Low Outliers: 1192, 1204

High Outliers: 1382

The Stem & Leaf diagram above illustrates the data collected for the iso-tank outputs. The

diagram was refined to omit outliers. Any value above the upper fence (1369.75) or below the

lower fence (1227.75) is considered an outlier. As shown, there were 3 outliers consisting of 2

unusually low values and one unusually high value. The calculations for the quartiles, inter-

quartile range, upper and lower fences are shown below:

Q2=[ n+12 ]

th

value¿ [ 67+12 ]th

value¿34 thvalue¿1298.75

Q1=[ n+14 ]

th

value¿ [ 67+14 ]th

value¿17th value¿1281

7 | P a g e

Q3=3 [ n+14 ]

th

value¿3[ 67+14 ]th

value¿51st value¿1316.5

IQ R=Q3−Q1¿1316.5−1281¿35.5

U pper Fence=Q3+1.5 (Q3−Q1 )¿1316.5+1.5 (1316.5−1281 )¿1369.75

L ower Fence=Q1−1.5 (Q3−Q1 )¿1281−1.5 (1316.5−1281 )¿1227.75

Below: Refined Box & Whiskers Plot Illustrating Output from Iso-Tanks

The

box-

plot

above

8 | P a g e

1400

1350

1300

1250

1200

Outp

ut /

Num

ber o

f 20

lb C

ylind

ers

Refined Boxplot of Output From Iso-Tanks

Q1 1281

Q2 1298.75

Q3 1316.5

Min 1229

Max 1335.75

IQR 35.5

illustrates the output of the iso-tanks. The five number summary is also shown in addition to

the inter-quartile range.

The box shows that the distribution is approximately symmetrical. In fact, it is perfectly

symmetrical since:

Q3−Q2 (17.75 )=Q2−Q1(17.75)

The box-plot above was refined to omit outliers. Any value above the upper fence (1369.75) or

below the lower fence (1227.75) is considered an outlier. As shown there were 3 outliers

consisting of 2 unusually low value and one unusually high value.

The minimum and maximum values refer to the value of the whiskers. If the box-plot was not

refined then the minimum value (lower whisker) would be 1192 whilst the maximum value

(upper whisker) would be 1382.

Below: Histogram Illustrating Number of 20 lb Cylinders Filled from Iso-Tanks

9 | P a g e

13901370135013301310129012701250123012101190

25

20

15

10

5

0

Mean 1294StDev 31.02N 67

Output From Iso-Tank (x) / No. of 20 lb Cylinders

Freq

uenc

y (y)

10

2

19

22

12

6

21

2

Normal Histogram Showing Output of Iso-Tanks

The histogram above shows the distribution of the number of filled 20 lb cylinders. The x-

values are the number of cylinders filled from an iso-tank, whilst the y-values are the

frequencies of each class.

The modal class (class with the highest frequency) is 1290 < x < 1310 as 22 iso-tanks produced

an output within this range. The second highest class is 1310 < x < 1330 with 19 iso-tanks, whilst

the third is 1270 < x < 1290 with 12 iso-tanks.

As shown by the distribution fit curve, the distribution of the output of iso -tanks is

approximately normal. This is seen better with the bars removed, as seen on the next page.

Below: Distribution Fit Curve for the Output of Iso-Tanks

10 | P a g e

13901370135013301310129012701250123012101190

18

16

14

12

10

8

6

4

2

0

Mean 1294StDev 31.02N 67

Output From Iso-Tank (x) / No. of 20 lb Cylinders

Freq

uenc

y (y)

1294

Normal Dsitribution Fit Curve of Iso-Tank Outputs

From the above curve, we can see that the distribution of the output from iso-tanks is

approximately normal. This is because:

The majority of the points are centered around the mean.

The distribution is approximately symmetrical.

The frequencies tail off fairly rapidly as the values of x move away from the mean.

The calculations for the mean and standard deviation are shown below:

x=∑ xn

¿ 86706.867 ¿1294.131343≈1294

σ=√ 1n−1 (∑ x2−

(∑ x )2

n )¿√ 167−1 (112273485−86706.8

2

67 )¿31.01745456≈31.02

11 | P a g e

Hypothesis Test

A 95% hypothesis test was performed to make a decision as to whether the mean output from

the iso-tanks ordered by Jn-Marie & Sons is actually 1300. The null and alternative hypotheses

are shown below:

H 0 : μ=1300

H 1: μ ≠1300

12 | P a g e

Rationale for Two-Tailed Test

As you can see, this is a two-tailed hypothesis test. According to Bruce E. Trumbo of the

Calfornia State University (CSU), two-sided alternatives should be used unless there is very

strong reason to support the use of a one-sided alternative. There was no compelling reason to

choose a one sided alternative hypothesis because the gas station had not expressed any

overwhelming beleief that they received way more or way less than the proposed amount of

output from the iso-tanks.

Results from Minitab

One-Sample Z

Test of μ = 1300 vs ≠ 1300The assumed standard deviation = 31.0175

N Mean SE Mean 95% CI Z P67 1294.13 3.79 (1286.70, 1301.56) -1.55 0.121

The above data shows the results from Minitab for a 95% Hypothesis Test.

As shown, the acceptance region is 1286.70 < µ < 1301.56 The mean of the data set (1294), falls

within this acceptance region, thus we retain H0.

Therefore, we are 95% certain that the mean output from the Iso-Tanks ordered by Jn-Marie

& Sons is equal to 1300.

Using the p-Value

We can also come to a conclusion using the p – value. Since the p – value (0.121) is greater than

α (0.05), we retain the null hypothesis.

We can also use the p-value to come to a conclusion for other confidence levels. Once the p-

value is greater than α then the null hypothesis is retained. Hence, let us investigate what

would be the conclusion at another common confidence level; 90%.

13 | P a g e

At the 90% confidence level, α=0.100. Hence we will retain the null hypothesis since the p-

value 0.121 > α (0.1). In fact at any confidence level higher than 90, H0 would be accepted, since

α gets smaller for higher confidence level.

Confidence Interval

The acceptance region calculated by Minitab also represents a 95% confidence interval for the

mean output iso-tanks of a sample size of 67. Therefore, we can conclude that if other samples

of a similar size were drawn, then 95% of them would have a mean output that falls between

the interval, (1286.70, 1301.56).

The calculations for the hypothesis test are shown below:

Finding critical value in terms of z

α=1−0.95¿0.05

1−α2=1−0.05

2 ¿0.975

c=Φ (0.975 )¿1.960

Therefore, the 95% acceptance region is -1.96 < z < 1.96.

Converting acceptance region from z to x values

This is the same thing as finding a 95% confidence interval so the confidence interval

formula was used:

CI=x± z σ√n

¿1294.131±1.96 31.01745456√67

¿1294.131±7.427¿ [1286.703 ,1301.558 ]

¿1286.70≤ x ≤1301.56

14 | P a g e

Analysis

Distribution and Skewness

The distribution of iso-tank outputs was found to be normally distributed; no significant skew in

the data was identified. This was seen in the distribution fit curve as the shape largely

represented a symmetrical bell curve whilst it was also shown in the refined box & whiskers

plot as Q3 – Q2 = Q2 – Q1.

The mean iso-tank output of 1294 is lower than the calculated median value of 1298.75. When

the outliers are omitted and a new mean is calculated, we get a trimmed mean of 1296. This

mean though still slightly less is closer to the median than the original mean. The difference

between the trimmed mean and the median (2.75) is extremely small and is not a large enough

difference to conclude that the distribution is negatively skewed.

One way to test the significance of this difference is Pearson’s 2nd Co-efficient of Skewness. This

statistic measures how much negatively or positively skewed that a distribution is. The co-

efficient lies between -3 and +3; highly negative values indicate a negative skewness whilst

highly positive values indicate positive skewness. A co-efficient of 0 means that a distribution

has no skewness at all. The calculated co-efficient of skewness is -0.45 which is much closer to 0

than it is to negative three. Hence, it can be concluded that though the distribution of iso-tanks

has a slight negative skew, it is not sufficiently significant to say that the distribution is

15 | P a g e

negatively skewed. The distribution is much closer to symmetrical than it is to a negatively

skewed one. The calculation for the co-efficient of skewness is shown below:

sk=3 ( x−Q2 )

σ¿3 (1294.131343−1298.75 )

31.01745456≈−0.45

Hypothesis Test – Is the mean iso-tank output actually equal to 1300?

As stated above, the mean iso-tank output is 1294. Hence, at first glance it may seem that the

gas station does not get the right amount of gas from the iso-tanks but a lower amount.

However, we have to keep in mind that the data collected is only a sample and the mean of

1294 is only an unbiased estimate of the population mean. Therefore a hypothesis test was

conducted to decide with 95% certainty, whether or not the mean iso-tank output is actually

equal to 1300.

The result of the hypothesis test showed that there was not enough evidence to conclude that

the mean iso-tank output was not 1300. Hence, with 95% certainty, it can be concluded that Jn-

Marie Gas Station receives the correct amount of gas from its iso-tanks.

The p-value was also used to test whether or not the null hypothesis would be retained at other

common levels of significance. It revealed that at any confidence level above 90%, the null

hypothesis would hold. This further strengthens the conclusion that the gas station does in fact

receive the correct volume of gas from its iso-tanks.

Confidence Interval

A 95% confidence interval was calculated for the mean output of an iso-tank and it was found

that for any sample of a similar size (67), 95% of them would yield a mean between 1286 and

1301 filled 20 lb cylinders.

Outliers

16 | P a g e

There were three outliers identified. The first two outliers were extremely low values (1192 ans

1204). According to Mr. JnMarie, the most likely reason for these low outliers is leaks that

would have occurred on the line between the iso-tank and the filling apparatus.

The other outlier was an extremely high value of 1382 filled 20 lb cylinders. This specific tank

was bought in the month of December. Mr. Jn-Marie said that this high yield is due to the need

for customers during the Christmas season to avoid running out of product. Rather than wait

for their tanks to empty before replenishment they hand in tanks that still have product. Hence,

it would take less product from the iso-tank to fill those tanks, allowing it to fill more than the

usual number of tanks; hence the apparent high yield.

17 | P a g e

Findings

1. The output of iso-tanks ordered by JnMarie & Sons is approximately normally

distributed.

2. It is 95% certain that the Jn-Marie Gas Station receives the correct amount of LPG from

its orders of iso-tanks.

3. Ninety-Five percent of samples consisting of 67 iso-tanks will produce a mean output

between 1286 and 1301 filled 20 lb cylinders.

4. Three outliers were found in the data collected. The two extremely low values are most

likely the results of leakages on the line between the tank and the filling apparatus,

whilst the high outlier is most likely due to customers returning tanks that were not

empty during the Christmas season.

18 | P a g e

Conclusion

We can conclude with 95% certainty that Jn-Marie & Sons gas station receives the correct

amount of LPG from its orders of iso-tanks.

19 | P a g e

Bibliography

Steve Dobbs, J. M. (2010). Advanced Level Mathematic : Statistics 1. Cambridge: Cambridge

University Press.

Steve Dobbs, J. M. (2011). Advanced Level Mathematics : Statistics 2. Cambridge: Cambridge

University Press.

J. Crawshaw, J. Chambers. (2001). A Concise Course in Advanced Level Statistics. Cheltenham,

UK: Nelson Thornes Ltd.

Bruce E. Trumbo. (1999). Confidence Intervals and Tests of Hypothesis: A Minitab

Demonstration. California: California State University, Hayward.

20 | P a g e