Financial Risk Manager Part 1

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A hospital uses ultrasound technology to measure the weight of unborn babies as follows:

Gestation period in weeks	30	32	34	36	38	40
Estimated weight of foetus	1.6	1.7	2.5	2.8	3.2	3.5

Further information: $S_{XX} = 70$ , $S_{YY} = 3.015$ , $S_{XY} = 14.3$ . Calculate the least square estimator of the slope and the Y-intercept (in that order)._

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TTanishq

Least square estimator: 0.2043; Y-intercept: -4.6

Least square estimator: 0.20; Y-intercept: -4

Least square estimator: 2.55; Y-intercept: 35

Least square estimator: 0.2043; Y-intercept: 35

Explanation:

Explanation

Under OLS (Ordinary Least Squares) estimation:

The regression equation is: $y_i = \alpha + \beta x_i$
Where Y is the dependent variable (foetal weight)
X is the independent variable (gestation period)
$\alpha$ = the y-intercept
$\beta$ = the slope

Calculating the slope ( $\beta$ ):

$\beta = \frac{S_{XY}}{S_{XX}} = \frac{14.3}{70} = 0.2043$

Calculating the y-intercept ( $\alpha$ ):

First, we need the means:

Mean of Y ( $\bar{y}$ ): $\bar{y} = \frac{(1.6 + 1.7 + 2.5 + 2.8 + 3.2 + 3.5)}{6} = 2.55$
Mean of X ( $\bar{x}$ ): $\bar{x} = \frac{(30 + 32 + 34 + 36 + 38 + 40)}{6} = 35$

Then: $\alpha = \bar{y} - \beta \bar{x} = 2.55 - 0.2043 \times 35 = -4.60$

Therefore, the correct values are:

Least square estimator (slope): 0.2043
Y-intercept: -4.6

This corresponds to option A.

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