Introduction
Simplify put, life expectancy has been said to be, ‘the number of years that an individual is expected to live as determined by statistics’ (Wilkinson, 1996, p.171; Banca, 2003, p.23). The years an individual could live are determined by actuaries. The calculations involving life expectancy are conducted during child births or other preferred times as desired. on a general note, it is expected that at birth, humans have the tendency to live up to eighty years, when as an individual who is 40 years could possibly live up to eightyfive years of age. The benefits accrued to projecting the expectations for how long individuals could live include the determination of the period s expected of individuals to attain retirement form work, and how long they continue to benefit from emoluments. Based on this notion, Belsley noted that:
‘Annuities payable for lifetimes are usually based on separate male or female tables, except that a qualified plan or trust must use unisex tables’ (Belsley, 1991, p.82).
For countries, the wide recognition is that the expectations for longevity of an individual are primarily determined by individual’s financial state of been, as well as healthiness but most determining among these are the negative aspects of poorhealth and poverty, which are quite known to interrelate (UNDP, 2003). It is however noted a misperception of the two as said:
‘…based on the experience of the last few generations, the assumption that rising prosperity remains the main source of improved standards of health throughout society constitutes some form of misconception’ (UNDP, 2003, p.64).
The usual believe is good health will continually follow a comfortable standard of living. But against this believe, it is noted that:
‘The misconception is that the only remaining relationship between income and health is the residual relationship between absolute poverty and illhealth’ (WorldBank, 2003, p.229).
These counteracting misconceptions have necessitated the need for a study on the actual facts that determine the longevity of individuals in a clearly defined statistical, comparative and computational manner. This paper, therefore, is a direct response through statistical determination of human longevity as applied to by factors that affect how long humans live. information on (i) Life expectancy at birth (years), (ii) Literacy Rate (% of people ages 15 and above), (iii) Infrastructure (% of rural population with access to water) and (iv) Gross National Income per capita from a random selection of thirty countries was obtained and imputed into a new excel spreadsheet, as well further analysis was conducted using eViews.
Methodology
This analysis was conducted using eViews on the following parameter:
 A (30 X 1) matrix Y represented the data on Life Expectancy; and
 A (30 X 4) matrix X as follows:

 The first (30 X 1) column contained only the number 1;
 The second (30 X 1) column represented the data on Literacy Rate;
 The third (30 X 1) column represented the data on Infrastructure; and
 The fourth (30 X 1) column represented the data on Gross National Income per capita
Solutions to Questions
Calculation of the matrix (X’X)^{1}X’Y
Supposing Y’s value is taken FROM Ymean, anD AN effort is made towards finding out the possibility OF Any similarity in X’s deviation from Xmean causes a form of linearity, no mater how awkward the relationship of the linearity may occur, the variance of X – Xmean may not certainly be explained by the variation in Y – Ymean; as a matter of fact, there could be a reasonably huge inexplicable difference which could constitute a huge estimation error. If this is the situation, the totaled squared errors could possibly be huge as well. Conversely to actualize an appealing fit, however with the imputation of numeric values, the ratio is then denoted by Explainablechange in Y/Total change in Y. This ratio is captured by the term R^{2} through the following:
 The term in the numerator, being only a part of the one in denominator, is the smaller of the two, and so R^{2} is never greater than 1;
 When there is hardly any linear relationship between X and Y, R^{2} is low (close to 0) – on the other hand, a good linearity is an indicator of any change in the value of Y inferring that the numerator is nearer the denominator and R^{2} is nearer to 1; and
 R^{2} = ( r_{XY}) ^{2}
The following page contains a covariance/correlation/regression result from the age, GDP data (where R^{2} is 0.3899, and suggests a poor fit):
Using EViews, the correlation coefficient r = 0.6244 between life expectancy and GDP is thus found:
It should be noted that r^{2} = 0.389923 = R^{2}
Regression
Hypothesis Testing
The BLUE property of the OLS estimates is used to test hypotheses about α and β; this involves testing the null hypothesis H_{o}: β = 0. Given a population regression equation;
Y = α + β X +Hε
The calculation of tStatistic in this case is usually referred to as a tratio due to the fact that it is the ratio of the OLS estimator to its estimated standard error
In the present situation, β can not be zero; hence it could be either positive or negative because a priori, the direction of causality is not obvious
Therefore test is conducted of
H_{o}: β = 0 against H_{1}: β ≠ 0
Because h_{0 }in the present case is sufficiently greater/smaller than zero, a twotail test will be used (using a 0.05 level of significant, the critical tvalue, with 3 d.o.f., is t_{0.05} = 3.18). The decision criterion here is Reject H_{0 }if tStatistic > 2.353
Given b = 19.46454 and s_{b} = 14.05681, tStatistic = b/s_{b} = 19.46454/14.05681= 1.384707. The tStatistic therefore is 1.384707, which is less than 2.353; hence reject H_{o} at the 0.05 level of significance is inappropriate (i.e. there is no evidence that Literacy Rate affects life expectancy. The following diagram implies:
Multivariate Regression Model
y = β_{0 n }+ β1×1 + β2×2 + β3×3 +… + ε
y is the DEPENDENT variable where as each of the x_{j} is an INDEPENDENT variable then, the following conditions are stated:
Each explanatory variable Xj is assumed:
(1A) to be deterministic or nonrandom; (1B): to come from a ‘fixed’ population; and (1C): to have a variance V(xj) which is not ‘too large’
Assumptions concerning the random term β_{0, }β_{1, }β_{2, }β_{3.}
(IIA) E(ε_{i}) = 0 for all I; (IIB) Var(ε_{i}) = σ^{2 }constant for all; IIC) Covariance (ε_{i, }ε_{k}_{ k}) = σ^{2} for any i and k
(IID) each of ε_{i} has a normal distribution
Properties of b_{0 }, b_{1 }, b_{2} , b_{3 :}
 Each of these statistics is a linear function of the Y values;
 Therefore, they all have normal distributions;
 Each is an unbiased estimator [i.e, E(b_{k}) = b_{0}_{]} b_{1 }and b_{2}
 Each b_{k} is the most efficient estimator of all unbiased estimators
Thus, each of b_{0 }, b_{1 }, b_{2} ….is:
Best; linear; unbiased; and estimator of the respective parameter
Conclusion
Each estimator b_{i} has a normal distribution with mean = β_{1}_{ bi}^{2 }and variance = s_{bi}^{2 }where s_{bi}^{2 }is unknown
The Test (at 10% Significance) of the Hypothesis that ‘
Infrastructure has a positive effect on Life Expectancy at birth’
The proposed regression model is:
Life expectancy = ß_{0} + ß_{1}(Literacy Rate) +ß_{2}(Infrastructure)+ ß_{3}(Gross National Income per capita)
It is hereby proposed that Life expectancy is the variable dependent on three independent variables:
Literacy Rate; Infrastructure; and Gross National Income per capita
The estimation of the proposed model is
Y^{e} = 1001.86 + 8.84* Infrastructure + 95.17* Literacy Rate + 1.51* Gross National Income per capita
Here Y^{e} is the estimated value of Life expectancy
The leastsquares estimates of the ßvalues are denoted by bvalues; thus, b_{1 }is the estimate of ß_{1} and b_{2 }is the estimate of ß_{2} (in this present case, b_{1 }= 8.84, and b_{2 }= 95.17
Gross National Income Per Capita has an effect on Life Expectancy at Birth
Step 1: Set up of the hypotheses
H_{0:} ß_{2 }= 0 (Gross National Income per capita has no effect)
H_{1:} ß_{2 }> 0(Gross National Income per capita has a positive effect)
Onetailed test.
Step 2: Selection of statistic
The estimator b_{2} is the teststatistic
Step3: Identification of the distribution of b_{2}
_{The above assumptions imply that b2 is }Best, and linear in the dependent variable income
Unbiased Estimator of b_{2}
Since b_{2} is unbiased, E(b_{2}) = b_{2}
b_{2} has a normal distribution because it is linear in Income
Thus, b_{2}~ N(b_{2}, b_{2}^{2}) where b_{2}^{2 }is unknown.
Step 4: Construction of test statistic
_{THE} use OF the standard error of b_{2 is employed here }because the true value of b_{2}^{2 }is unknown
Therefore, the test statistic is t≡b (b_{2}– b_{2}) / (standard error of b_{2}) has a tdistribution with 204 = 16 d.o.f.
As b_{2} = 0 under the null hypothesis (H0)t = b2 / (standard error of b_{2})
Considering the regression output, it can be seen that b2 = 95.16913 and standard error of b2 = 38.53613. EViews therefore produces a tstatistic regarding Gross National Income per capit of 2.46907 (= 95.16914/38.53614.
The corresponding probability value is 0.0251.
Since this is a onetail test, the half the probability value is accepted
^{S}tep 5: Comparing with critical value t^{C}
t^{C} = 1.336756 for a onetailed test with significance level () = 0.1 and d.o.f. = 16
t^{C} = 1.336756 < 2.469606
Step 6: Conclusion drawn
The test is significant as follow: Rejection of H0 at 10% and at 5% (1.745883 < 2.469606) but not at 1% (2.583492 > 2.469607)
Step 7: Interpretation of result
It can be said that the data has been supported (with nearly 98% accuracy), hence the hypothesis that Gross National Income Per Capita has an effect on Life Expectancy at Birth is accepted.
Reference List
Banca, D., 2003. Survey on Italian Households Wealth and Income (SIHWI). Rome: Banca d’Italia.
Belsley, D., 1991. Conditioning diagnostics: collinearity and weak data in regression. New York: Wiley.
UNDP., 2003. Human Development Report 2003. New York: Oxford University Press.
Wilkinson, R., 1996. Unhealthy societies: the affliction of inequalities. London: Routledge.
WorldBank., 2003. World Development Indicators, 2003. Washington, DC: World Bank.