The weights to the nearest kilogram, of a group of 50 students in a College of Technology are given below:
65, 70, 60, 46, 51, 55, 59, 63, 68, 53, 47, 53, 72, 53, 67, 62, 64, 70, 57, 56, 73, 56, 48, 51, 58, 63, 65, 62, 49, 64, 53, 59, 63, 50, 48, 72, 67, 56, 61, 64, 66, 52, 49, 62, 71, 58, 53, 69, 63, 59.
(a) Prepare a grouped fraquency table with class intervals 45 - 49, 50 - 54, 55 - 59 etc.
(b) Using an assumed mean of 62 or otherwise, calculate the mean and standard deviation of the grouped data, correct to one decimal place.
65, 70, 60, 46, 51, 55, 59, 63, 68, 53, 47, 53, 72, 53, 67, 62, 64, 70, 57, 56, 73, 56, 48, 51, 58, 63, 65, 62, 49, 64, 53, 59, 63, 50, 48, 72, 67, 56, 61, 64, 66, 52, 49, 62, 71, 58, 53, 69, 63, 59.
(a) Prepare a grouped fraquency table with class intervals 45 - 49, 50 - 54, 55 - 59 etc.
(b) Using an assumed mean of 62 or otherwise, calculate the mean and standard deviation of the grouped data, correct to one decimal place.
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Correct Answer: Option n
Explanation:
(a)
(b)
\(Mean (\bar{x}) = A + \frac{\sum f(x - A)}{\sum f}\)
= \(62 + \frac{-135}{50}\)
= \(62 - 2.7 = 49.3\)
Standard deviation = \(\sqrt{\frac{\sum f(x - A)}{\sum f}}\)
= \(\sqrt{\frac{3275}{50}}\)
= \(\sqrt{65.5}\)
= \(8.093 \approxeq 8.1\) (to 1 decimal place)
(a)
| Class Interval | Tally | Freq |
| 45 - 49 | |||| | | 6 |
| 50 - 54 | |||| |||| | 9 |
| 55 - 59 | |||| |||| | 10 |
| 60 - 64 | |||| |||| || | 12 |
| 65 - 69 | |||| || | 7 |
| 70 - 74 | |||| | | 6 |
(b)
| ClassInterval | Mid-value(x) | x - 62 | \((x - 62)^{2}\) | \(f\) | \(f(x - 62)\) | \(f(x - 62)^{2}\) |
| 45 - 49 | 47 | -15 | 225 | 6 | -90 | 1350 |
| 50 - 54 | 52 | -10 | 100 | 9 | -90 | 900 |
| 55 - 59 | 57 | -5 | 25 | 10 | -50 | 250 |
| 60 - 64 | 62 | 0 | 0 | 12 | 0 | 0 |
| 65 - 69 | 67 | 5 | 25 | 7 | 35 | 175 |
| 70 - 74 | 72 | 10 | 100 | 6 | 60 | 600 |
| \(\sum\) | 50 | -135 | 3275 |
\(Mean (\bar{x}) = A + \frac{\sum f(x - A)}{\sum f}\)
= \(62 + \frac{-135}{50}\)
= \(62 - 2.7 = 49.3\)
Standard deviation = \(\sqrt{\frac{\sum f(x - A)}{\sum f}}\)
= \(\sqrt{\frac{3275}{50}}\)
= \(\sqrt{65.5}\)
= \(8.093 \approxeq 8.1\) (to 1 decimal place)