Answer:
a) 4.5
b) x = 11.2, s = 4.65
c) 93.33%
Step-by-step explanation:
We are given he following data in the question:
8, 12, 11, 15, 14, 10, 8, 3, 8, 7, 21, 12, 9, 19, 11
a) Estimation of standard deviation using range
Sorted data: 3, 7, 8, 8, 8, 9, 10, 11, 11, 12, 12, 14, 15, 19, 21
Range = Maximum - Minimum = 21 - 3 = 18
Range rule thumb:
[tex]s = \dfrac{\text{Range}}{4} = \dfrac{18}{4} = 4.5[/tex]
b) Mean and standard deviation
Formula:
[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]
where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean =\displaystyle\frac{168}{15} = 11.2[/tex]
Sum of squares of differences = 302.4
[tex]S.D = \sqrt{\dfrac{302.4}{14}} = 4.65[/tex]
c) fraction of the scores actually lie in the interval x ± 2s
[tex]x \pm 2s = 11.2 \pm 2(4.65) = (1.9,20.5)[/tex]
Since 14 out of 15 entries lie in this range, we can calculate the percentage as,
[tex]\dfrac{14}{15}\times 100\% = 93.33\%[/tex]
