( a ) Determine the values of sample average and standard deviation of the data in Problem 19.11.
( b ) Determine the values 1 and 2 standard deviations from the mean. Of the 50 sample points, how many fall within these two ranges?
Question : ( a ) Determine the values of sample average and standard de...
(a) Convert P(X) data to frequency values to determines.
| X | P(X) | XP(X) | f | { X }^{ 2 } | { fX }^{ 2 } |
| 1 | 0.2 | 0.2 | 10 | 1 | 1 |
| 2 | 0.2 | 0.4 | 10 | 4 | 40 |
| 3 | 0.2 | 0.6 | 10 | 9 | 90 |
| 6 | 0.1 | 0.6 | 5 | 36 | 180 |
| 9 | 0.2 | 1.8 | 10 | 81 | 810 |
| 10 | 0.1 | 1 | 5 | 100 | 500 |
| 4.6 | 1630 |
Sample average: \overline { X } = 4.6
Sample variance: { s }^{ 2 }=\frac { 1630 }{ 49 } -\frac { 50 }{ 49 } { \left( 4.6 \right) }^{ 2 }=11.67
Std deviations = { (11.67) }^{ 0.5 } = 3.42
(b) \overline { X } \pm 1s is 4.6 \pm 3.42 = 1.18 and 8.02
25 values, or 50%, are in this range.
\overline { X } \pm 2s is 4.6 \pm 6.84 = –2.24 and 11.44
All 50 values, or 100%, are in this range.