Surveying Multiple Choice Questions on “Survey Adjustments and Errors Theory – General Principles of Least Squares”.
1. The principle of least squares can be formed from___________
a) Probability equation
b) Normal equation
c) Celestial equation
d) Observed equation
Answer: a
Clarification: Though the principle of least squares is useful in the determination of certain parameters it is actually derived from the probability equation. A probability equation shows an accidental error with definite law.
2. In the principle of least squares, residual error will be__________
a) Maximum
b) Minimum
c) Negligible
d) Nor error occurs
Answer: b
Clarification: The general method of least squares is used in case of determination of normal equations and the laws of weight. It consists of a set of observations, which are having a minimum residual error. Residual error is the difference in the adopted value and observed value.
3. Which of the following indicates the formula of arithmetic mean?
a) ∑V * n
b) ∑V – n
c) ∑V + n
d) ∑V / n
Answer: d
Clarification: The formula of arithmetic mean can be given as ∑V / n, where n indicates the number of observed values and V indicates difference of the individual with the mean of the readings.
4. If the value of number of observations is large and e is small then ∑e / n will be equal to __________
a) Frequency
b) True value
c) Arithmetic mean
d) Probable value
Answer: c
Clarification: The value of number of observations and the e values are to be maintained in a stable manner. If the n is increased and e is decreased, the value ∑e / n will be infinitesimally small and will be equal to arithmetic mean.
5. Which of the following indicates the fundamental of least squares?
a) arithmetic mean should be maximized
b) arithmetic mean should be zero
c) arithmetic mean should be neutralized
d) arithmetic mean should be minimized
Answer: d
Clarification: The fundamental of law of least squares include the sum of squares of the residuals found by the use of arithmetic mean should be minimized. This must be applied while using the principle of least squares.
6. Which of the following value must always be positive in the principle of least squares?
a) (N-(∑V/n))2
b) (N*(∑V/n))2
c) (N+(∑V/n))2
d) (N-(∑V/n))0
Answer: a
Clarification: The term must always be positive because it eliminates the possible error which will occur during the process of recording values. It can also be believed that the presence of square can make it positive.
7. In the process of least squares, sum of residuals must be equal to zero.
a) False
b) True
Answer: b
Clarification: While determining the relation between the different set of values considered, we assume certain preferences which results in sum of residuals being zero.
8. When number of observations is large, the arithmetic mean will be equal to true value.
a) True
b) False
Answer: a
Clarification: It is known that the increase in number of observations leads to increase in the value of e. this can lead to equality of mean and the true value. This is the key point used in the principle of least squares.
9. Determine the arithmetic mean if the summation of the difference between individual and mean of the observations is given as 1.54 for 10 observations.
a) 0.154
b) 0.514
c) 0.145
d) 1.405
Answer: a
Clarification: The arithmetic mean can be calculated by using the formula,
M = ∑V / n. On substitution, we get
M = 1.54 / 10
M = 0.154.
10. Find the value of N if the sums of squares of two individual residuals are given as 24.98 and 13.76 respectively. The summation of the difference between individual and the mean of the readings is 3.32 for 8 observations.
a) 1
b) 1.6
c) 1.7
d) 2
Answer: b
Clarification: By applying the principle of least squares the value of N can be determined by,
∑rꞌ2 = ∑r2 + n*(N-(∑V/n))2. On substitution, we get
24.98 = 13.76 + 8*(N-(3.32/8))2
N = 1.6.