Advertisements
Advertisements
Question
The mean and variance of seven observations are 8 and 16 respectively. If five of these are 2, 4, 10, 12 and 14, then find the remaining two observations.
Solution
Let the missing two observations be ‘a’ and ‘b’
Arithmetic mean = 8
| xi | xi2 |
| 2 | 4 |
| 4 | 16 |
| 10 | 100 |
| 12 | 144 |
| 14 | 196 |
| a | a2 |
| b | b2 |
| `sumx_"i"` = 42 + a + b | `sumx_"i"^2` = 460 + a2 + b2 |
`(2 + 4 + 10 + 12 + 14 + "a" + "b")/7` = 8
⇒ `(42 + "a" + "b")/7` = 8
a + b + 42 = 56
a + b = 56 − 42
a + b = 14 ...(1)
variance = 16
variance = `(sumx_"i"^2)/"n" - ((sumx_"i")/"n")^2`
16 = `(460 + "a"^2 + "b"^2)/7 - (8^2)`
⇒ 16 = `(460 + "a"^2 + "b"^2)/7 - 64`
16 + 64 = `(460 + "a"^2 + "b"^2)/7`
⇒ 80 = `(460 + "a"^2 + "b"^2)/7`
560 − 460 = a2 + b2
a2 + b2 = 100
⇒ (a + b)2 − 2ab = 100 ...[a2 + b2 = (a + b)2 − 2ab]
142 − 2ab = 100
⇒ 196 − 2ab = 100 ...[a + b = 14 (from (1)]
196 − 100 = 2ab
96 = 2ab
⇒ ab = `96/2` = 48
∴ b = `48/"a"` ...(2)
Substitute the value of b = `48/"a"` in (1)
`"a" + 48/"a"` = 14
⇒ a2 + 48 = 14a
a2 − 14a + 48 = 0
⇒ (a − 6) (a − 8) = 0
a = 6 or 8
|
When a = 6 b = `48/"a"` = `48/6` = 8 |
When a = 8 b = `48/"a"` = `48/8` = 6 |
∴ Missing observation is 8 and 6 or 6 and 8
APPEARS IN
RELATED QUESTIONS
If the range and the smallest value of a set of data are 36.8 and 13.4 respectively, then find the largest value
Calculate the range of the following data.
| Income | 400 − 450 | 450 − 500 | 500 − 550 | 550 − 600 | 600 − 650 |
| Number of workers | 8 | 12 | 30 | 21 | 6 |
A teacher asked the students to complete 60 pages of a record notebook. Eight students have completed only 32, 35, 37, 30, 33, 36, 35 and 37 pages. Find the standard deviation of the pages yet to be completed by them.
Find the standard deviation of the first 21 natural numbers.
The rainfall recorded in various places of five districts in a week are given below. Find its standard deviation
| Rainfall (in mm) | 45 | 50 | 55 | 60 | 65 | 70 |
| Number of places | 5 | 13 | 4 | 9 | 5 | 4 |
The measurements of the diameters (in cms) of the plates prepared in a factory are given below. Find the standard deviation
| Diameter (cm) | 21 − 24 | 25 − 28 | 29 − 32 | 33 − 36 | 37 − 40 | 41 − 44 |
| Number of plates | 15 | 18 | 20 | 16 | 8 | 7 |
The sum of all deviations of the data from its mean is
The mean of the following frequency distribution is 62.8 and the sum of all frequencies is 50. Compute the missing frequencies f1 and f2.
| Class Interval | 0 − 20 | 20 − 40 | 40 − 60 | 60 − 80 | 80 − 100 | 100 − 120 |
| Frequency | 5 | f1 | 10 | f2 | 7 | 8 |
The diameter of circles (in mm) drawn in the design are given below.
| Diameters | 33 − 36 | 37 − 40 | 41 − 44 | 45 − 48 | 49 − 52 |
| Number of circles | 15 | 17 | 21 | 22 | 25 |
Calculate the standard deviation.
If for a distribution, `sum(x - 5)` = 3, `sum(x - 5)^2` = 43, and total number of observations is 18, find the mean and standard deviation
