Number Systems

The least square method is used to solve over-determined systems of equations. The given system of equations is: We can write these equations in matrix form: To solve this using the least square method, we calculate the normal equation: Where: First, calculate : Next, calculate : Now, solve for : Using matrix inversion or substitution, we find: Thus, the most probable values of and are and .

The regression equations are:
From the equations, we can extract the values of (slope of the regression of on ) and (slope of the regression of on ):
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- The correlation coefficient can be calculated using the formula:
Substituting the values:
Thus, the correlation coefficient between and is approximately when rounded to 2 decimal places.

The formula for the Chi-square statistic is: Where:
- is the sample size
- is the sample variance
- is the population variance
Given that:
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- Degrees of freedom implies , hence
Now calculate the Chi-square value: Final Answer:

The most probable value of a quantity is the value that best represents the true value in terms of measurements. It may not always be equal to the true value but is typically the value that is closest to the true value based on observed data.
Step 1: Understanding the context of the question. The most probable value is essentially the value that comes closest to the true value, even if it's not exactly equal. In many measurements, errors or variations occur, but the most probable value is usually the one that minimizes these errors.
Step 2: Evaluating the options. - Option (A) is incorrect because the most probable value does not always increase with an increase in the true value. It depends on the distribution of the observed values.
- Option (B) is also incorrect because the most probable value doesn’t always decrease with a decrease in the true value; it’s based on how the data is distributed.
- Option (C) is incorrect because the most probable value is not necessarily equal to the true value, as measurement errors and uncertainties exist.
- Option (D) is correct because the most probable value is typically the one that is nearest to the true value, as it reflects the most consistent and repeated measurements.
Thus, the correct answer is (D).