The following are the measured values of the angle P with their r
The following are the measured values of the angle P with their respective weights
I. 55° 25’ 17’’, weight 4
II. 55° 25’ 22’’, weight 3
III. 55° 25’ 26’’, weight 2
The probable error of the mean will beA. – 5.33’’
B. 1. 33’’
C. – 3.67’’
D. 1.71’’
Please scroll down to see the correct answer and solution guide.
Right Answer is: D
SOLUTION
Concept:
Probable error of the mean (Em)w
\({({{\rm{E}}_{\rm{m}}})_{\rm{w}}} = \pm 0.6745\sqrt {\frac{{\sum \left( {{\rm{w}}{{\rm{v}}^2}} \right)}}{{\left( {{\rm{n}} - 1} \right)\sum {\rm{w}}}}}\)
v = Measured value – Most probable value
w = weight
n = no. of observations
Calculation:
The weighted average of seconds in each angle is
\(\frac{{4 \times 17 + 3 \times 22 + 2 \times 26}}{9} = \frac{{186}}{9} = 20.67\)
∴ Residuals
17 - 20.67 = -3.67
22 - 20.67 = 0.67
26 - 20.67 = 5.33
|
S.No |
Value |
Weight |
Value × Weight (seconds) |
v (seconds) |
V2 |
wv2 |
|
1 |
55° 25’ 17’’ |
4 |
68 |
– 3.67’’ |
13.47 |
53.88 |
|
2 |
55° 25’ 22’’ |
3 |
66 |
1. 33’’ |
1.768 |
5.304 |
|
3 |
55° 25’ 26’’ |
2 |
65 |
5.33’’ |
28.41 |
56.82 |
|
|
|
9 |
189 |
|
|
116.00 |
\({\rm{Most\;probable\;value}} = 55^\circ {25^{\rm{'}}}{\left[ {\frac{{17 \times 4 + 22 \times 3 + 26 \times 2}}{{4 + 3 + 2}}} \right]^{{\rm{''}}}}\)
Most probable value = 55° 25’ 20.67’’
v1 = - 3.67’’
v2 = 1.33’’
v3 = 5.33’’
\({({{\rm{E}}_{\rm{m}}})_{\rm{w}}} = \pm 0.6745\left[ {\sqrt {\frac{{116}}{{\left( {3 - 1} \right) \times 9}}} } \right]\)
\({({{\rm{E}}_{\rm{m}}})_{\rm{w}}} = \pm 1.71{\rm{''}}\)