Advertisements
Advertisements
प्रश्न
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.
उत्तर
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
संबंधित प्रश्न
Find the range and coefficient of range of the following data
43.5, 13.6, 18.9, 38.4, 61.4, 29.8
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 |
Find the variance and standard deviation of the wages of 9 workers given below:
₹ 310, ₹ 290, ₹ 320, ₹ 280, ₹ 300, ₹ 290, ₹ 320, ₹ 310, ₹ 280
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 |
In a study about viral fever, the number of people affected in a town were noted as Find its standard deviation
| Age in years | 0−10 | 10−20 | 20−30 | 30−40 | 40−50 | 50−60 | 60−70 |
| Number of people affected | 3 | 5 | 16 | 18 | 12 | 7 | 4 |
The range of the data 8, 8, 8, 8, 8 ........ 8 is
Variance of first 20 natural numbers 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 |
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
If the range and coefficient of range of the data are 20 and 0.2 respectively, then find the largest and smallest values of the data
