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Question
The frequency distribution is given below.
| x | k | 2k | 3k | 4k | 5k | 6k |
| f | 2 | 1 | 1 | 1 | 1 | 1 |
In the table, k is a positive integer, has a variance of 160. Determine the value of k.
Solution
Assumed mean = 3k
| x | fi | d = xi − A = xi − 3k |
fidi | fidi2 |
| k | 2 | − 2k | − 4k | 8k2 |
| 2k | 1 | − k | − k | k2 |
| 3k | 1 | 0 | 0 | 0 |
| 4k | 1 | k | k | k2 |
| 5k | 1 | 2k | 2k | 4k2 |
| 6k | 1 | 3k | 3k | 9k2 |
| `sumf_"i"` = 7 | `sumf_"i""d"_"i"` = k | `sumf_"i""d"_"i"^2` = 23k2 |
Here `sumf_"i"` = 7, `sumf_"i""d"_"i"` = k, `sumf_"i""d"_"i"^2` = 23k2
Variance = 160
`(sumf_"i""d"_"i"^2)/(sumf_"i") - ((sumf_"i""d"_"i")/(sumf_"i"))^2` = 160
`(23"k"^2)/7 - ("k"/7)^2` = 160
`(23"k"^2)/7 - ("k"^2/49)` = 160
⇒ `(161"k"^2 - "k"^2)/49` = 160
`(160"k"^2)/49` = 160
⇒ 160k2 = 160 × 49
k2 = `(160 xx 49)/160`
k = `sqrt(49)`
⇒ k = 7
The value of k = 7
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