Comparison Of Food And Feed Production In Australia And Other Countries – A Statistical Study

Research Questions

We know that the world population is continuously growing and demand for food for this population is also increases. So, it is necessary to increase the food production for providing the food for each one. Also, animal population also increasing and feed production for animal should be increase. For this project, we have to compare the food and feed production for the Australia and two countries with highest production of food and feed. We have to compare the variation pattern in the use of food and feed for the different countries. For this purpose, we have to collect the data and then we have to check some hypotheses by using proper statistical tools and techniques. Let us see this statistical data analysis in detail. 

Research Questions

For this research study, the research questions are summarised as below:

1. Which countries produce most production of food and feed?
2. Is there a statistically significant difference exists between the average production of feed and food for Australia?
3. Is there a statistically significant difference exists between the average production of feed and food for India?
4. Is there a statistically significant difference exists between the average production of feed and food for USA?
5. Are there statistically significant differences in the average feed and food production for the three countries such as Australia, India, and USA?

Data Collection

For this research project, data is collected from The Food and Agriculture Organization of the United Nations. United Nations Organization provides the free access for this data for 245 countries and territories from the year 1961 to 2013. Required website for this data is given in the reference list.

Descriptive Statistics

We know that the descriptive statistics provides us the idea about the nature of variables included in the data set.

Here, we have to see some descriptive statistics for the Rice feed and food production in world. All numbers in the following tables are given in thousand tonnes. First of all we have to see the descriptive statistics for the Rice feed and food production for the world. Descriptive statistics for feed and food production for entire world is summarised as below:

Y2013
Rice (Food) production for world in ‘000 tones
Mean	2168.310345
Standard Error	834.9732259
Median	107
Mode	1
Standard Deviation	11014.0533
Sample Variance	121309370.1
Kurtosis	70.21168333
Skewness	8.117025439
Range	108321
Minimum	0
Maximum	108321
Sum	377286
Count	174

From above descriptive statistics, it is observed that the average rice food production for the year 2013 is given as 2168 thousand tones with the standard deviation of 11014 thousand tones. The maximum rice food production for single country in 2013 is observed as 108321 thousand tones.

Now, we have to see the descriptive statistics for the rice feed production for all countries which is given as below:

Y2013
Rice (Feed) World Production in ‘000 tones
Mean	414.7407407
Standard Error	182.7726754
Median	5
Mode	0
Standard Deviation	1644.954078
Sample Variance	2705873.919
Kurtosis	36.46249072
Skewness	5.805432137
Range	12052
Minimum	0
Maximum	12052
Sum	33594
Count	81

The average rice feed production for world is given as 415 thousand tonnes with the standard deviation of 1645 thousand tonnes. The maximum rice feed production is given as 12052 thousand tonnes.

The list of first ten countries in the production of rice food is given as below:

Food

Area	Y1961	Y1971	Y1981	Y1991	Y2001	Y2011	Y2012	Y2013
China, mainland	26687	58217	74814	86891	100658	109404	107956	108321
India	30617	37289	45355	68068	74649	86478	86933	87006
Indonesia	7669	12855	18896	23616	27046	32431	33175	33637
Bangladesh	8761	10178	12493	17583	23295	26387	26681	26892
Viet Nam	4903	6374	7424	9006	12412	13066	12865	13253
Philippines	2465	3231	4724	5373	8256	11290	11689	11752
Thailand	4023	6060	6494	6725	7115	7461	7654	7677
Japan	10529	9365	8442	7983	7450	6641	7469	7609
Myanmar	2410	2992	4120	5171	6441	6548	6844	7073
Brazil	2869	3790	4925	5989	6266	6727	6309	6437

 It is observed that China is placed at top in the rice food production.

Data Collection

The list of first ten countries for the rice feed production is given as below:

Feed

Area	Y1961	Y1971	Y1981	Y1991	Y2001	Y2011	Y2012	Y2013
China, mainland	2869	6137	8978	10008	15342	11589	12100	12052
Myanmar	228	990	2405	2136	4119	7786	7538	7882
Viet Nam	66	79	116	446	1302	2526	2589	2712
Thailand	339	458	593	408	1122	2410	2499	2405
India	143	172	213	1495	1866	2106	2105	2124
Indonesia	176	334	540	1089	1552	1879	1873	1832
Bangladesh	192	199	273	363	726	1014	1011	1031
Iran	0	0	0	79	200	534	534	534
Nepal	76	90	118	153	277	392	464	457
Philippines	73	112	153	321	432	364	394	402

It is observed that China is placed top for the rice feed production.

Now, we have to see some descriptive statistics for the feed and food production in Australia, India, and United States of America. All numbers in the following tables are given in thousand tonnes. First of all we have to see the descriptive statistics for the feed and food production for the country Australia. Descriptive statistics for feed and food production for Australia is summarised as below:

Descriptive Summary
Australia
	Feed	Food
Mean	684.1111111	435.3960396
Median	97	75
Mode	2	0
Minimum	0	0
Maximum	7936	5474
Range	7936	5474
Variance	2673752.1026	885667.6016
Standard Deviation	1635.1612	941.0991
Coeff. of Variation	239.02%	216.15%
Skewness	3.7778	3.6598
Kurtosis	15.7139	15.5990
Count	27	101
Standard Error	314.6869	93.6429

 From above table, it is observed that the average feed production for Australia is given as 684.11 thousand tonnes with the standard deviation of 1635.16 thousand tonnes. The average food production for Australia is given as 435.40 thousand tonnes with the standard deviation of 941.10 thousand tonnes.

Now, we have to see these descriptive statistics for the feed and food production for the country India which is given as below:

Descriptive Summary
India
	Feed	Food
Mean	3779.153846	11466.06481
Median	312	1111.5
Mode	0	0
Minimum	0	0
Maximum	22757	185884
Range	22757	185884
Variance	40572128.1354	814513849.7434
Standard Deviation	6369.6254	28539.6890
Coeff. of Variation	168.55%	248.91%
Skewness	2.2656	3.7386
Kurtosis	4.8132	15.7770
Count	26	108
Standard Error	1249.1863	2746.2329

 From above table, it is observed that the average feed production for India is given as 3779.15 thousand tonnes with the standard deviation of 6369.63 thousand tonnes. The average food production for India is given as 11466.06 thousand tonnes with the standard deviation of 28539.69 thousand tonnes.

Now, we have to see this descriptive statistics for the food and feed production for the country USA which is given as below:

Descriptive Summary
USA
	Feed	Food
Mean	8246.194444	6112.152381
Median	374.5	892
Mode	165	0
Minimum	0	0
Maximum	140096	81513
Range	140096	81513
Variance	961442432.4468	185683111.3612
Standard Deviation	31007.1352	13626.5590
Coeff. of Variation	376.02%	222.94%
Skewness	4.0547	3.8044
Kurtosis	15.3711	17.1086
Count	36	105
Standard Error	5167.8559	1329.8160

From above table, it is observed that the average feed production for USA is given as 8246.194444 thousand tonnes with the standard deviation of 31007.1352 thousand tonnes. The average food production for USA is given as 6112.152381 thousand tonnes with the standard deviation of 13626.5590 thousand tonnes.

Graphical Analysis

Graphical analysis plays an important role in easy understanding of the concepts. Here, we have to use the graphical analysis for showing the variation of the production in different countries. In this topic, we have to see some graphical analysis of the given data set. Most dark blue colour in the following graph shows the highest yield or production by country.

There are total 115 distinct items included in this study. From big data analysis, it is observed that the highest product of the feed and food in the world is produced within two countries such as USA and India. Now, we have to see some box plots for the comparison of the feed and food for the three countries Australia, India, and USA. Box plots for the food and feed for the Australia are given as below:Australia food and feed production box plot

From above boxplots it is observed that there are outliers exist in the data for the feed and food production in Australia.

Now, we have to see the box plots for the feed and food production in India. Box plots for the feed and food production in India are summarised as below:India feed and food production box plots

Descriptive Statistics

 From the above box plots, it is observed that there are outliers exist in the data for the feed and food production in India.

Now, we have to see the same comparison for the USA, that is we have to see the box plots for the feed and food production in United States of America. Required box plots are given as below:USA feed and food production box plotsFrom above box plots, it is observed that there are outliers exist in the data for the feed and food production in United States of America.

Independent Samples t-test

In this section, we have to use the independent samples t tests for checking the significant difference between the two population means. Here, we have to use this two sample t test for checking the statistically significant difference in the average production between feed and food for all countries. Also, we have to check three claims for comparison purpose. First of all we have to check whether the average Rice food production is more or equal to average rice feed production or not. The null and alternative hypothesis for this test is given as below:

Null hypothesis: H₀: The average Rice food production is more or equal to average rice feed production.

Alternative hypothesis: H_a: The average Rice food production is less than average rice feed production.

H₀: µ₁ ≥ µ₂ versus H_a: µ₁ < µ₂

This is a one tailed test.

Here, we consider 5% level of significance for this two sample t test.

The software output for this test is summarised as below:

Calculations Area
Pop. 1 Sample Variance	121308196.0000
Pop. 2 Sample Variance	2706025.0000
Pop. 1 Sample Var./Sample Size	697173.5402
Pop. 2 Sample Var./Sample Size	33407.7160
For one-tailed tests:
TDIST value	0.0208
1-TDIST value	0.9792

Separate-Variances t Test for the Difference Between Two Means
(assumes unequal population variances)
Data
Hypothesized Difference	0
Level of Significance	0.05
Population 1 Sample
Sample Size	174
Sample Mean	2168
Sample Standard Deviation	11014.0000
Population 2 Sample
Sample Size	81
Sample Mean	415
Sample Standard Deviation	1645.0000
Intermediate Calculations
Numerator of Degrees of Freedom	533748972026.5930
Denominator of Degrees of Freedom	2823493979.4644
Total Degrees of Freedom	189.0385
Degrees of Freedom	189
Standard Error	854.7405
Difference in Sample Means	1753.0000
Separate-Variance t Test Statistic	2.0509
Lower-Tail Test
Lower Critical Value	-1.6530
p-Value	0.9792
Do not reject the null hypothesis

We do not reject the null hypothesis that the average Rice food production is more or equal to average rice feed production. There is insufficient evidence to conclude that the average Rice food production is less than average rice feed production.

Now, we have to check another hypothesis or claim which is given as below:

Null hypothesis: H₀: There is no any statistically significant difference exists between the average production of feed and food for Australia.

Alternative hypothesis: H_a: There is a statistically significant difference exists between the average production of feed and food for Australia.

H₀: µ₁ = µ₂ versus H_a: µ₁ ≠ µ₂

This is a two tailed test.

Here, we consider 5% level of significance for this two sample t test.

The software output for this test is summarised as below:

Australia

Group Statistics
	Element	N	Mean	Std. Deviation	Std. Error Mean
Y2013	Feed	27	684.1111	1635.16119	314.68692
Food	101	435.3960	941.09915	93.64286

Independent Samples Test
	t-test for Equality of Means
t	df	Sig. (2-tailed)	Mean Difference	Std. Error Difference	95% Confidence Interval of the Difference
Lower	Upper
Y2013	Equal variances assumed	1.025	126	.307	248.71507	242.67304	-231.52774	728.95789
Equal variances not assumed	.758	30.746	.455	248.71507	328.32429	-421.13118	918.56132

For this test, the p-value is given as 0.455 (equal variances not assumed) which is greater than the level of significance or alpha value 0.05, so we do not reject the null hypothesis that There is no any statistically significant difference exists between the average production of feed and food for Australia.

There is insufficient evidence to conclude that there is a statistically significant difference exists between the average production of feed and food for Australia.

Graphical Analysis

Now, we have to check same hypothesis for India. Here, we have to check whether there is a statistically significant difference exists between the average production of feed and food for India or not. We have to use same test i.e. two sample t test for checking this hypothesis or claim. The null and alternative hypothesis for this test is given as below:

Null hypothesis: H₀: There is no any statistically significant difference exists between the average production of feed and food for India.

Alternative hypothesis: H_a: There is a statistically significant difference exists between the average production of feed and food for India.

H₀: µ₁ = µ₂ versus H_a: µ₁ ≠ µ₂

This is a two tailed test.

Here, we consider 5% level of significance for this two sample t test.

The software output for this test is summarised as below:

Group Statistics
	Element1	N	Mean	Std. Deviation	Std. Error Mean
Ye2013	Feed	26	3779.1538	6369.62543	1249.18632
Food	108	11466.0648	28539.68903	2746.23286

Independent Samples Test
	t-test for Equality of Means
t	df	Sig. (2-tailed)	Mean Difference	Std. Error Difference	95% Confidence Interval of the Difference
Lower	Upper
Ye2013	Equal variances assumed	-1.362	132	.176	-7686.91097	5645.73665	-18854.73609	3480.91415
Equal variances not assumed	-2.548	131.723	.012	-7686.91097	3016.99542	-13654.94207	-1718.87986

For this test, the p-value is given as 0.012 (equal variances not assumed) which is less than the level of significance or alpha value 0.05, so we reject the null hypothesis that there is no any statistically significant difference exists between the average production of feed and food for India.

There is sufficient evidence to conclude that there is a statistically significant difference exists between the average production of feed and food for India.Now, we have to check same hypothesis for USA. Here, we have to check whether there is a statistically significant difference exists between the average production of feed and food for USA or not. We have to use same test i.e. two sample t test for checking this hypothesis or claim. The null and alternative hypothesis for this test is given as below:

Null hypothesis: H₀: There is no any statistically significant difference exists between the average production of feed and food for USA.

Alternative hypothesis: H_a: There is a statistically significant difference exists between the average production of feed and food for USA.

H₀: µ₁ = µ₂ versus H_a: µ₁ ≠ µ₂

This is a two tailed test.

Here, we consider 5% level of significance for this two sample t test.

The software output for this test is summarised as below:

USA 

Group Statistics
	Element2	N	Mean	Std. Deviation	Std. Error Mean
Yr2013	Feed	36	8246.1944	31007.13519	5167.85586
Food	105	6112.1524	13626.55904	1329.81600

Independent Samples Test
	t-test for Equality of Means
t	df	Sig. (2-tailed)	Mean Difference	Std. Error Difference	95% Confidence Interval of the Difference
Lower	Upper
Yr2013	Equal variances assumed	.566	139	.572	2134.04206	3769.95801	-5319.83493	9587.91905
Equal variances not assumed	.400	39.730	.691	2134.04206	5336.21072	-8653.12609	12921.21022

For this test, the p-value is given as 0.691 (equal variances not assumed) which is greater than the level of significance or alpha value 0.05, so we do not reject the null hypothesis that there is no any statistically significant difference exists between the average production of feed and food for USA.

Independent Samples t-test

There is insufficient evidence to conclude that there is a statistically significant difference exists between the average production of feed and food for USA.

Two Way ANOVA

In this section, we have to see some advanced analysis for the given data. Here, we want to check whether average food and feed production is same for the three nations Australia, India, and United States of America. For checking this hypothesis we have to use two analysis of variance or ANOVA. The null and alternative hypothesis for this test is given as below:Null hypothesis: H₀: There are no any statistically significant differences in the average feed and food production for the three countries such as Australia, India, and USA. 

Alternative hypothesis: H_a: There are statistically significant differences in the average feed and food production for the three countries such as Australia, India, and USA. 

Here, we consider 5% level of significance for this two sample t test.

The software output for this test is summarised as below:

Between-Subjects Factors
	N
Area	Australia	128
India	134
United States of America	141
Element	Feed	89
Food	314

Descriptive Statistics
Dependent Variable:Y2013
Area	Element	Mean	Std. Deviation	N
Australia	Feed	684.1111	1635.16119	27
Food	435.3960	941.09915	101
Total	487.8594	1120.32829	128
India	Feed	3779.1538	6369.62543	26
Food	11466.0648	28539.68903	108
Total	9974.5746	25927.24058	134
United States of America	Feed	8246.1944	31007.13519	36
Food	6112.1524	13626.55904	105
Total	6657.0142	19472.25468	141
Total	Feed	4647.1011	20125.81460	89
Food	6127.6624	18992.43598	314
Total	5800.6898	19232.75348	403

Required ANOVA table is summarised as below:

Tests of Between-Subjects Effects
Dependent Variable:Y2013
Source	Type III Sum of Squares	df	Mean Square	F	Sig.
Corrected Model	7.412E9	5	1.482E9	4.166	.001
Intercept	7.153E9	1	7.153E9	20.099	.000
Area	2.745E9	2	1.373E9	3.857	.022
Element	2.132E8	1	2.132E8	.599	.439
Area * Element	1.221E9	2	6.105E8	1.715	.181
Error	1.413E11	397	3.559E8
Total	1.623E11	403
Corrected Total	1.487E11	402
a. R Squared = .050 (Adjusted R Squared = .038)

For entire model, the significance value or the P-value is given as 0.001 which is less than alpha value 0.05, so we reject the null hypothesis that there are no any statistically significant differences in the average feed and food production for the three countries such as Australia, India, and USA. There is sufficient evidence to conclude that there are statistically significant differences in the average feed and food production for the three countries such as Australia, India, and USA.

Estimated marginal means are summarised as below:

1. Grand Mean
Dependent Variable:Y2013
Mean	Std. Error	95% Confidence Interval
Lower Bound	Upper Bound
5120.512	1142.158	2875.078	7365.947

Estimates
Dependent Variable:Y2013
Area	Mean	Std. Error	95% Confidence Interval
Lower Bound	Upper Bound
Australia	559.754	2043.563	-3457.805	4577.312
India	7622.609	2060.533	3571.689	11673.529
United States of America	7179.173	1821.752	3597.686	10760.661

Pairwise comparisons for this study are given as below

Pairwise Comparisons
Dependent Variable:Y2013
(I) Area	(J) Area	Mean Difference (I-J)	Std. Error	Sig.a	95% Confidence Interval for Differencea
Lower Bound	Upper Bound
Australia	India	-7062.856*	2902.059	0.015	-12768.2	-1357.53
United States of America	-6619.420*	2737.688	0.016	-12001.6	-1237.24
India	Australia	7062.856*	2902.059	0.015	1357.531	12768.18
United States of America	443.436	2750.378	0.872	-4963.69	5850.561
United States of America	Australia	6619.420*	2737.688	0.016	1237.243	12001.6
India	-443.436	2750.378	0.872	-5850.56	4963.69
Based on estimated marginal means
*. The mean difference is significant at the .05 level.
a. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).

Univariate Tests
Dependent Variable:Y2013
	Sum of Squares	df	Mean Square	F	Sig.
Contrast	2.745E9	2	1.373E9	3.857	.022
Error	1.413E11	397	3.559E8
The F tests the effect of Area. This test is based on the linearly independent pairwise comparisons among the estimated marginal means.

1. Element

Estimates
Dependent Variable:Y2013
Element	Mean	Std. Error	95% Confidence Interval
Lower Bound	Upper Bound
Feed	4236.486	2020.854	263.574	8209.399
Food	6004.538	1065.013	3910.768	8098.307

Pairwise Comparisons
Dependent Variable:Y2013
(I) Element	(J) Element	Mean Difference (I-J)	Std. Error	Sig.a	95% Confidence Interval for Differencea
Lower Bound	Upper Bound
Feed	Food	-1768.051	2284.316	.439	-6258.920	2722.818
Food	Feed	1768.051	2284.316	.439	-2722.818	6258.920

Univariate Tests
Dependent Variable:Y2013
	Sum of Squares	df	Mean Square	F	Sig.
Contrast	2.132E8	1	2.132E8	.599	.439
Error	1.413E11	397	3.559E8
The F tests the effect of Element. This test is based on the linearly independent pairwise comparisons among the estimated marginal means.

4. Area * Element
Dependent Variable:Y2013
Area	Element	Mean	Std. Error	95% Confidence Interval
Lower Bound	Upper Bound
Australia	Feed	684.111	3630.562	-6453.418	7821.641
Food	435.396	1877.133	-3254.967	4125.759
India	Feed	3779.154	3699.721	-3494.341	11052.649
Food	11466.065	1815.281	7897.300	15034.830
United States of America	Feed	8246.194	3144.159	2064.913	14427.476
Food	6112.152	1841.031	2492.764	9731.540

 Advance Analysis

Total Rice Food production in year 2013 for entire world is given as 377286000 tones.

We consider total food consumption is same as total food production. (Most of the times estimate for total production and total consumptions are approximately same or near. For Convenience, we consider that total consumption is same as total production, we don’t have proper estimates available for total consumption. There is lack of data for some countries.)

Estimated world Population for year 2013 is 7,213,426,452.

(This population estimate for world population is taken from internet 

Now, we have to find out the per capita food consumption for the year 2013.

Per capita food consumption = Total food consumption / Total population

Per capita food consumption = 377286000 / 7213426452 tones per person per year

Per capita food consumption = 377286000 / 7213426452 tones per person per year 

Per capita food consumption = 0.0523 tones per person per year

Per capita food consumption = 52.3 kg/year

(Daily 143 grams of rice food consumption per person) 

From this analysis, it is observed that per capita rice food consumption for the year 2013 is given as 52.3 kg. 

Conclusions

For this research study, conclusions are summarised as below:

1. It is observed that the average rice food production for the year 2013 is given as 2168 thousand tones with the standard deviation of 11014 thousand tones. The maximum rice food production for single country in 2013 is observed as 108321 thousand tones.
2. The average rice feed production for world is given as 415 thousand tonnes with the standard deviation of 1645 thousand tonnes. The maximum rice feed production is given as 12052 thousand tonnes.
3. It is observed that China is placed at top in the rice food and feed production.
4. There is insufficient evidence to conclude that the average Rice food production is less than average rice feed production.
5. It is observed that the average feed production for Australia is given as 684.11 thousand tonnes with the standard deviation of 1635.16 thousand tonnes. The average food production for Australia is given as 435.40 thousand tonnes with the standard deviation of 941.10 thousand tonnes.
6. It is observed that the average feed production for India is given as 3779.15 thousand tonnes with the standard deviation of 6369.63 thousand tonnes. The average food production for India is given as 11466.06 thousand tonnes with the standard deviation of 28539.69 thousand tonnes.
7. It is observed that the average feed production for USA is given as 8246.194444 thousand tonnes with the standard deviation of 31007.1352 thousand tonnes. The average food production for USA is given as 6112.152381 thousand tonnes with the standard deviation of 13626.5590 thousand tonnes.
8. It is observed that the highest product of the feed and food in the world is produced within two countries such as USA and India.
9. It is observed that there are outliers exist in the data for the feed and food production in United States of America, India, and Australia.
10. There is insufficient evidence to conclude that there is a statistically significant difference exists between the average production of feed and food for Australia.
11. There is sufficient evidence to conclude that there is a statistically significant difference exists between the average production of feed and food for India.
12. There is insufficient evidence to conclude that there is a statistically significant difference exists between the average production of feed and food for USA.
13. There is sufficient evidence to conclude that there are statistically significant differences in the average feed and food production for the three countries such as Australia, India, and USA.
14. It is observed that per capita rice food consumption for the year 2013 is given as 52.3 kg. 

Recommendations for CEO

From above data analytics study, some recommendations for the CEO are made in terms of product description. These recommendations are given as below:

1. It is observed that the production of rice in both categories i.e. food and feed is not uniformly distributed all over the world and it is concentrated in four to five countries. So, there would be big scope for increasing the production of rice in other countries.
2. Apart from China and India, Australia has a chance to increase the yield of rice by implementing some plans.
3. In the view of investing, growing countries should be considered and variation pattern would be kept in mind.
4. The study of variation pattern and different environmental conditions suggests that there is a variation in quality and quantity.

Cover Letter

To,

CEO,

XYZ company,

(Address)

Hello,

With regarding the data analytics for the given big data for the item rice food and feed, this study revealed some facts which are summarised below:

1. It is observed that the average rice food production for the year 2013 is given as 2168 thousand tones with the standard deviation of 11014 thousand tones. The maximum rice food production for single country in 2013 is observed as 108321 thousand tones.
2. The average rice feed production for world is given as 415 thousand tonnes with the standard deviation of 1645 thousand tonnes. The maximum rice feed production is given as 12052 thousand tonnes.
3. It is observed that China is placed at top in the rice food and feed production.
4. There is insufficient evidence to conclude that the average Rice food production is less than average rice feed production.
5. It is observed that the average feed production for Australia is given as 684.11 thousand tonnes with the standard deviation of 1635.16 thousand tonnes. The average food production for Australia is given as 435.40 thousand tonnes with the standard deviation of 941.10 thousand tonnes.

Some of the recommendations based on this study are summarised as below:

1. It is observed that the production of rice in both categories i.e. food and feed is not uniformly distributed all over the world and it is concentrated in four to five countries. So, there would be big scope for increasing the production of rice in other countries.
2. Apart from China and India, Australia has a chance to increase the yield of rice by implementing some plans.
3. In the view of investing, growing countries should be considered and variation pattern would be kept in mind.
4. The study of variation pattern and different environmental conditions suggests that there is a variation in quality and quantity.

Hope, these study outputs will help you in further decision making. Thank you                                                                          

Mr/Ms ABC

Sr. Analyst,

(Date and sign) 

References

Casella, G. and Berger, R. L. (2002). Statistical Inference. Duxbury Press.

Cox, D. R. and Hinkley, D. V. (2000). Theoretical Statistics. Chapman and Hall Ltd.

Degroot, M. and Schervish, M. (2002). Probability and Statistics. Addison – Wesley.

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