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International Journal of Electrical and Electronics Engineering (IJEEE) ISSN(P): 2278-9944; ISSN(E): 2278-9952 Vol. 7, Issue 4, Jun – Jul 2018, 9-20 © IASET CONFIDENCE INTERVAL OF ANNUAL EXTREMUM OF AMBIENT AIR TEMPERATURE AT DHUBRI

Rinamani Sarmah Bordoloi 1 , Dhritikesh Chakrabarty 2 & Manash Pratim Kasyap 3 1 Research Scholar, Department of Statistics, Assam Down Town University, Panikhaiti, Guwahati Assam, India 2 Research Guide, Department of Statistics, Handique Girls’ College, Panikhaiti, Guwahati, Assam, India 3 Research Guide, Department of Statistics, Assam Down Town University, Panikhaiti, Guwahati, Assam, India

ABSTRACT

Confidence intervals (of 95%, 99% & 99.73% degrees of confidence) have been determined for each of annual maximum & annual minimum of ambient air temperature at Dhubri. The determination is based on the data from 1969 onwards collected from the Regional Meteorological Centre at Tezpur. This paper describes the method of determination of them and the numerical findings on them.

KEYWORDS: Ambient Air Temperature at Dhubri, Annual Extremum, Confidence Interval

Article History

Received: 21 May 2018 | Revised: 01 Jun 2018 | Accepted: 18 Jun 2018

I. INTRODUCTION

Observations or data collected from experiments or survey suffer from chance error (which is unavoidable or uncontrollable) even if all the assignable (or intentional) causes or the sources of errors are controlled or eliminated and consequently the findings obtained by analyzing the observations or data which are free from the assignable errors are also subject to errors due to the presence of chance error in the observations (D.Chakrabarty2014) (R. Sarmah Bordoloiand and D. Chakrabarty 2016). Determination of parameters, in different situations, based on the observations is also subject to error due to the same reason. Searching for mathematical models describing the association of a chance error with the observations is necessary for analyzing the errors. There are innumerable situations/forms corresponding to the scientific experiments. The simplest one is that where observations are composed of some parameter and chance errors (D.Chakrabarty2014, 2015,2008). The existing methods of estimation namely least squares method, maximum likelihood method, minimum variance unbiased method, method of moment and method of minimum chi-square (Ivory1825, G.A.Barnard1949, Lucien Le Cam, Birnbaum Allan 1962, Erich L. Lehmann 1990, Anders Hold 1999) provides the estimator of the parameter which suffers from some error. In other words, none of these methods can provide the true value of the parameter. However, An analytical method has been developed, by Chakrabarty [D.Chakrabarty2014] for determining the true value of the parameter from observed data in the situation where the observations consist of a single parameter chance error but any assignable error. The method has already been successfully applied in determining the central tendency of each of annual maximum and annual minimum of the ambient air temperature at Guwahati

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Rinamani Sarmah Bordoloi, Dhritikesh Chakrabarty & Manash Pratim Kasyap

(D.Chakrabarty2014).Again the method has already been successfully applied in determining the central tendency of each of annual maximum and annual minimum of the ambient air temperature at Tezpur (Ramani Sarmah Bordoloi 2016). This paper deals with the determination of the central tendency of each of annual maximum and annual minimum of the ambient air temperature at Tezpur by the same method. The study was carried out using the data since 1969 onwards.

2. GAUSSIAN DISCOVERY

In the year 1809, German mathematician Carl Friedrich Gauss discovered the most significant probability

distribution in the theory of statistics popularly known as normal distribution, the credit for which discovery is also given by some authors to a French mathematician Abraham De Moivre who published a paper in 1738 that showed the normal distribution as an approximation to the binomial distribution discovered by James Bernoulli in 1713 {Bernoulli (1713),

Chakrabarty (2005b, 2008), De Moivre (1711, 1718), Gauss (

), Kendall and Stuart (1977, 1979), Walker and Lev

(1965), Walker (1985), Brye (1995), Hazewinkel (2001), Marsagilia (2004), Stigler (1982), Weisstein (--------) et al}. The normal probability distribution plays the key role in the theory of statistics as well as in the application of statistics.

There are innumerable situations where one can think of applying the theory of normal probability distribution to handle the situations.

The probability density function of normal probability distribution discovered by Gauss is described by the probability density function

 f(x : µ, σ) = { σ (2ᴨ) ½}. exp [ ─ ½ {(x ─ µ)/σ} 2 ], (2.1) ─ ∞

Confidence Interval of Annual Extremum of Ambient Air Temperature at Dhubri

11

3. CONFIDENCE INTERVAL

If X 1 , X 2 , …………, X n are n observations on µ (some characteristic / measure / parameter whose value is to be determined).

In this situation, each observation X i is composed of the true value of µ and an error ε i (occurring due to chance).

Thus the observations, in such types of situations, satisfy the model

X i = T(µ) + ε i

where (i) X i is the i th observation on µ,

(ii) T (µ) is the true value of µ

& (iii) ε i is the chance error associated with X i .

Let X 1 , X 2 , …………, X n be, n observations on µ (some characteristic / measure / parameter whose value is to be determined).

T (µ), the true value µ is unique.

But the observed values on µ are different.

The variation in the observed values occurs due to two types of causes/errors namely

1. Assignable Cause(s) that is (are) avoidable / controllable

& 2. Chance, Cause/Error that is unavoidable / uncontrollable

The values X i (i = 1, 2, ………

,

n) should be constant if there exists no cause of variation among them over i.

However, the chance cause of variation exists always.

Thus, if no assignable cause of variation exists in X i (i = 1, 2, ………

X i = T(µ) + ε i , (i = 1, 2, ………

, n)

where (i) X i is the i th observation on µ,

(ii) T(µ) is the true value of µ

& (iii) ε i is the chance error associated to X i .

, n), we have

(i = 1, 2, 3, …………

).

Here ε 1 , ε 2, ………

,

It is to be noted that

ε n are values of the chance error variable ε associated to X 1 , X 2 , …………, X n respectively.

X 1 , X 2 , …………, X n are known,

T(µ), ε 1 , ε 2 , …………, ε n are unknown

The number of linear equations in (3.1) is n with n + 1 unknowns implying that the equations are not solvable mathematically.

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Rinamani Sarmah Bordoloi, Dhritikesh Chakrabarty & Manash Pratim Kasyap

Reasonable Facts Regarding ε i :

ε n are unknown values of the variables ε.

ε 1 , ε 2, ………

,

The values ε 1 , ε 2 , …………, ε n are very small relative to the values X 1 , X 2 , …………, X n .

The variable ε assumes both positive and negative values.

P(ε = a) = P(ε = a) for every a assumed by ε.

Sum of all possible values of each ε is 0 (zero).

Standard deviation of ε is unknown and small, say σ ε .

ε obeys the normal probability law. Thus ε ~ N(0, σ ε ).

3.1. Confidence Interval of Error ‘ε’:

Since ε ~ N(0, σ ε ),

By the area property of Gaussian distribution given by the equation (2.5),

P( 1.96 σ ε < ε < 1.,96 σ ε ) = 0.95

i.e. the interval

( 1.96 σ ε 1.96 σ ε )

is the 95% confidence interval of ε.

(3.1)

(3.2)

This means that out of 100 random observations on ε (unknown), maximum 5 observations fall outside this

interval.

Again by the area property of Gaussian distribution given by the equations (2.6) and (2.7),

P( 2.58 σ ε < ε < 2.58 σ ε ) = 0.99

& P( 3σ ε < ε < 3σ ε ) = 0.9973

Respectively which implies that the intervals

( 2.58 σ ε , 2.58 σ ε ) & ( 3σ ε , 3σ ε )

are respectively the 99% & 99.73% confidence intervals of ε.

(3.3)

(3.4)

(3.5)

(3.6)

These respectively mean that out of 100 random observations on ε (unknown), maximum 1 observation falls outside the interval ( 2.58 σ ε , 2.58 σ ε ) and out of 10000 random observations on ε (unknown), maximum 27 observations fall outside the interval

( 3σ ε , 3σ ε ).

3.2. Confidence Interval of Parameter ‘µ’:

Also under the assumption number (7),

Confidence Interval of Annual Extremum of Ambient Air Temperature at Dhubri

 X ─ µ ~ N(0, σ ε ) or equivalently X ~ N(µ, σ ε ).

13

Thus, by the same area property of Gaussian distribution mentioned above,

(i) P(X 1.96 σ ε < µ < X + 1.96 σ ε ) = 0.95,

` i.e. the interval

(X 1.96 σ ε , X + 1.96 σ ε ) (3.8)

is the 95% confidence interval .

(3.7)

This means that out of 100 random intervals corresponding to each random observation, the value of µ will fall outside a maximum 5 such intervals.

(ii) P(X 2.58 σ ε < µ < X + 2.58 σ ε ) = 0.99

i.e. the interval

(X 2.58 σ ε , X + 2.58 σ ε )

is the 99% confidence interval of µ.

(3.9)

(3.10)

This means that out of 100 random intervals corresponding to each random observation, the value of µ will fall outside a maximum 1 such intervals.

& (iii) P(X 3σ ε < µ < X + 3σ ε ) = 0.9973

\(3.11)

i.e. the interval

(X 3σ ε , X + 3σ ε )

is the 99.73% confidence interval of µ.

(3.12)

This means that out of 10000 random intervals corresponding to each random observation, the value of µ will fall outside a maximum 27 such intervals.

3.3. Confidence Interval of Observation Variable ‘X ’:

Also under the assumption number (7),

 X ─ µ ~ N(0, σ ε ) or equivalently X ~ N(µ, σ ε ).

Thus, by the same area property of Gaussian distribution mentioned above,

(i) P(µ ─ 1.96 σ ε < X < µ + 1.96 σ ε ) = 0.95,

i.e. the interval

(µ ─ 1.96 σ ε , µ + 1.96 σ ε )

is the 95% confidence interval of X.

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(3.13)

(3.14)

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Rinamani Sarmah Bordoloi, Dhritikesh Chakrabarty & Manash Pratim Kasyap

This means that out of 100 random observations, maximum 5 observations fall outside this interval.

(ii) P(µ ─ 2.58 σ ε < X < µ + 2.58 σ ε ) = 0.99

i.e. the interval

(µ ─ 2.58 σ ε , µ + 2.58 σ ε )

is the 99% confidence interval of X.

This means that out of 100 random observations, maximum 1 observations fall outside this interval.

& (iii) P(µ ─ 3σ ε < X < µ + 3σ ε ) = 0.9973

i.e. the interval

(µ─ 3σ ε , µ + 3σ ε )

is the 99.73% confidence interval of X.

This means that out of 10000 observations, maximum 27 observations fall outside the interval.

Note

The set of observations

X 1 ,X 2 , ………, X i , …… ,

X n

constitute the population for the period from the year ‘1’ to the year ‘n’.

Thus, µ = Arithmetic Mean of (X 1 , X 2 , ………, X i , ……

and σ ε 2 = Variance of (X 1 , X 2 , ……, X i , …

, X n )

Table 1

, X n )

(3.15)

(3.16)

(3.16)

(3.17)

(3.18)

(3.19)

 Year Observed Value (X i ) (X i − 36.3 ) 2 Year No Observed (X i −36.3 ) 2 No Value 1 36.5 0.04 13 35.8 0.25 2 36.1 0.04 14 35.3 1.00 3 36.2 0.01 15 35.8 0.25 4 35.2 1.21 16 35.5 0.64 5 39.6 10.89 17 35.3 1.00 6 35.7 0.36 18 36.4 0.01 7 37.8 2.25 19 36.8 0.25 8 38.4 4.41 20 35.2 1.21 9 35.7 0.36 21 36.3 00 10 35.1 1.44 22 35.5 0.64 11 38.7 5.76 23 35.0 1.69 12 37.5 1.44 24 36.2 0.01 25 36.0 0.09

=

= × 907.6

= 36.304

Confidence Interval of Annual Extremum of Ambient Air Temperature at Dhubri

Ϭ 2 = =

= × 35.25

= 1.41

( - ) 2

Ϭ = 1.874342087

Confidence interval for µ

95% confidence interval for µ

(

X -1.96

, X + 1.96

)

= ( 36.304 – 1.96× . , 36.304 + 1.96× . )

= ( 35.5692579019, 37.0387420981 )

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Therefore, at 95% confidence interval of µ the ambient temperature at Dhubri is

( 35.5692579019, 37.0387420981 )

99% confidence interval for µ

(

X -2.58×

, X + 2.58×

)

= ( 36.304 – 2.58× . , 36.304 + 2.58× . )

= ( 35.3368394832, 37.2711605168 )

Therefore, at 99% confidence interval of µ the ambient temperature at Dhubri is

( 35.3368394832, 37.2711605168 )

99.73% confidence interval for µ

(

X -3×

, X + 3×

)

= ( 36.304 – 3× .

, 36.304 + 3× .

)

= ( 35.1793947478, 37.4286052522)

Therefore, at 99.73% confidence interval of µ the ambient temperature at Dhubri is

( 35.1793947478, 37.4286052522)

CONFIDENCE INTERVAL OF X

95% Confidence interval for X

( µ

- 1.96 σ, µ + 1.96 σ )

= ( 36.3 – 1.96 × 1.874342087, 36.3 + 1.96 × 1.874342087 )

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Rinamani Sarmah Bordoloi, Dhritikesh Chakrabarty & Manash Pratim Kasyap

= ( 32.6262895095, 39.9737104905 )

Therefore at 95% confidence interval of X the ambient temperature at Dhubri is

( 32.6262895095, 39.9737104905 )

99% Confidence interval for X

( µ

- 2.58× σ, µ + 2.58× σ )

= ( 36.3 – 2.58 × 1.874342087, 36.3 + 2.58 × 1.874342087 )

= ( 31.4641974156, 41.1358025844 )

Therefore, at 99% confidence interval of X the ambient temperature at Dhubri is

99.73% Confidence interval for X

( µ

- 2.58× σ, µ + 2.58× σ )

= ( 36.3 – 3 × 1.874342087, 36.3 + 3 × 1.874342087 )

= ( 30.676973739, 41.923026261 )

Therefore, at 99.73% confidence interval of X the ambient temperature at Dhubri is

( 30.676973739, 41.923026261 )

5. ANALYSIS OF ANNUAL MINIMUM TEMPERATURE AT DHUBRI

Table 2

 Year No Observed Value (X i − 8.8 ) 2 Year No Observed Value (X i − 8.8 ) 2 1 8.1 0.49 13 6.1 7.29 2 7.3 2.25 14 10.0 1.44 3 8.8 00 15 9.5 0.49 4 9.2 0.16 16 10.5 2.89 5 9.3 0.25 17 8.9 0.01 6 9.6 0.64 18 9.1 0.09 7 8.6 0.04 19 12.2 11.56 8 7.6 1.44 20 10.0 1.44 9 8.9 0.01 21 9.0 0.04 10 8.4 0.16 22 7.8 1.00 11 9.4 0.36 23 5.8 9.00 12 8.8 00

=

= × 202.9

i

=

8.82173913043

Ϭ

2 =

( µ) 2

= × 41.05

= 1.78478260869

Confidence Interval of Annual Extremum of Ambient Air Temperature at Dhubri

Ϭ = √1.78478260869

= 1.33595756245

Confidence interval for µ

95% confidence interval of µ

( -1.96 , + 1.96

)

= (8.82173913043 – 1.96 × . # , 8.82173913043 + 1.96 × . #

= ( 8.27574897011, 9.36772929075)

)

17

Therefore, at 95% confidence interval of µ the ambient temperature at Dhubri is

( 8.27574897011, 9.36772929075)

99% confidence interval of µ

( -2.58 , + 2.58

)

= (8.82173913043 – 2.58 × . # , 8.82173913043 + 2.58 × . # )

= (8.10303779694, 9.54044046392 )

Therefore,

at

99%

9.54044046392)

confidence

interval

of

µ

the

ambient

temperature

99.73% confidence interval of µ

( -2.58 , + 2.58

)

= (8.82173913043 – 3 × . # , 8.82173913043 + 3 × . # )

= ( 7.98603990544, 9.65743835542)

at

Dhubri

is

(8.10303779694,

Therefore, at 99.73% confidence interval of µ the ambient temperature at Dhubri is

( 7.98603990544, 9.65743835542)

Confidence interval for X

95% confidence interval of X

( µ

- 1.96 σ, µ + 1.96 σ )

= ( 8.8 – 1.96 × 1.33595756245, 8.8 – 1.96 × 1.33595756245 )

= ( 6.1815231776, 11.4184768224 )

Therefore, at 95% confidence interval of X the ambient temperature at Dhubri is

( 6.1815231776, 11.4184768224 )

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Rinamani Sarmah Bordoloi, Dhritikesh Chakrabarty & Manash Pratim Kasyap

99% confidence interval of X

( µ

- 2.58 σ, µ + 2.58 σ )

= ( 8.8 – 2.58 × 1.33595756245, 8.8 + 2.58 × 1.33595756245 )

= ( 5.35322948888,12.2467705111 )

Therefore, at 99% confidence interval of X the ambient temperature at Dhubri is

( 5.35322948888,12.2467705111 )

99.73% confidence interval of X

( µ - 3 σ, µ + 3 σ )

= ( 8.8 – 3 × 1.33595756245, 8.8 + 3 × 1.33595756245 )

= ( 4.79212731265, 12.8078726873 )

Therefore, at 99.73% confidence interval of X the ambient temperature at Dhubri is

( 4.79212731265, 12.8078726873 )

6. CONCLUSIONS

The existing statistical methods of estimation yield estimates which are not free from error.

However, the method developed by Chakra artist  can yield the estimate which is free from error (i.e. exactly equal to the true value of the parameter). Thus the central tendency of the annual maximum as well as an annual minimum of the ambient air temperature at Dibrugarh as the available data yield, can be taken as 36.3 Degree Celsius and 8.8 Degrees Celsius respectively. Based on these two central tendency Confidence intervals (of 95%, 99% & 99.73% degrees of confidence) have been determined for each of annual maximum & annual minimum of ambient air temperature in Dibrugarh.

The determination of these two is based on the assumption that the data recorded by the Indian Meteorological Department have been recorded correctly. If there is an error in recording the data, the determined value(s) will not be accurate. The determination of these two is based on another assumption that the change in temperature at Dibrugarh during the period whose data have been used in computation has not been influenced by any assignable cause(s). If in this period, some assignable cause has influenced significantly on the change in temperature at this location, the findings are bound to be inaccurate.

7. REFERENCES

1. Ivory, “On the Method of Least Squares”, Phil. Mag., vol. LXV, pp. 3 – 10, 1825.

2. G. A. Barnard, “Statistical Inference”, Journal of the Royal Statistical Society (Series B), vol. 11, pp. 115 – 149,

1949.

3. Birnbaum Allan, “On the Foundations of Statistical Inference”, Journal of the American Statistical Association, vol. 57, pp. 269 – 306, 1962.

Confidence Interval of Annual Extremum of Ambient Air Temperature at Dhubri

19

4. Lucien Le Cam, “Maximum Likelihood — An Introduction”, ISI Review, vol. 58, no. 2, pp. 153 –171, 1990.

5. Erich L. Lehmann & George Casella, “Theory of Point Estimation”, Springer. ISBN 0 – 387 – 98502 – 6, 1998.

6. Anders Hald, “On the History of Maximum Likelihood in Relation to Inverse Probability an d Least Squares”, Statistical Science, vol. 14, pp. 214 – 222, 1999.

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11. Sharma, Gr, Et Al. "Ambient Temperature Trend Analysis For The North Saurashtra Region In View Of Climate Change."

12. Dhritikesh Chakrabarty (2014) : “ Temperature in Assam: Natural Extreme Value ”, J. Chem. Bio. Phy. Sci. (E- ISSN : 2249 – 1929), Sec. C, 4 (2), 1479 –1488. Online available at: www.jcbsc.org.

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Rinamani Sarmah Bordoloi, Dhritikesh Chakrabarty & Manash Pratim Kasyap

19. Dhritikesh Chakrabarty (2015 – 16) : “Confidence Interval of Annual Extremum of Ambientm Air Temperature at Guwahati”, J. Chem. Bio. Phy. Sci. (E- ISSN : 2249 – 1929), Sec. C, 6(1), 192 – 203. Online available at:

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23. Rinamani Sarmah Bordoloi and D. Chakrabarty “CONFIDENCE INTERVAL OF ANNUAL EXTREMUM OFAMBIENT AIR TEMPERATURE AT GUWAHATI “Journal of Mathematics & Systems Sciences (JMASS) Vol. 12, No. 1-2, 2016, ISSN : 0975-5454

24. Rinamani Sarmah Bordoloi and D. Chakrabarty “Determination of Parameter from ObservationContaining Itself and Chance Error: Central Tendency of Annual Extremum of Ambient airTemperature at Dibrugarh” International Journal of Advanced Research in Science, Engineering and Technology, ISSN: 2350-0328 Vol. 3, Issue 8, August 2016
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