Question

Example 13: Use Chebyshev's theorem with X = 68, n = 85 and S = 10 and do questions a- (a) Calculate the percentage (p) of data points that is within k =3 standard deviations of the mean. Substitute the given value of k into Chebyshev's formula and evaluate p • Write p as a percentage to one decimal place. p88.9% (b) Find the number of standard deviation (k) on either side of the mean that cuts off p=75% of the data. • First convert 75% to a decimal. • • Then substitute the decimal value of p into the formula p Solve the equation for k. Remember that k>1 k = 2 (c) Find the interval on which at least 75% of the data lies. • • The interval on which at least the proportion p values exist is given by: X uks Use the correct value of k calculated in part 13(b) above Use the mean X = 68 and the standard deviation S = 10, given above The lower limit must be written first in interval notation • Interval ( 48 88 )

Answer 1

To use Chebyshev's Theorem, let us state first, gg a stardom variable has mean pe and Standard deviation & then for any positive integes ki PUIX-Me i Ro) she P8 IX-MK Roz z 1-4 de : R2 Here we have given mean Standard eleviation = 10 Sample size = 10 is (a) When k=3, the probability of data points within R=3 Standard deviation of the means is equal to P{1-1112 Bony > 1 - 1 p$ x-pel € Bary i = 0.888 = 0.899 The the percentage = 0.8997100 = 88.9 % (6) Here given that the lorobability that the p{1x-pulskoy = 0 75 = 3 P$ 1x-M1 < Raya 1 2 3 4 1ů RP

and 3) on ... ill K2 - 4 R = 2 PS I X-elskay El-1 =3 R2 4 When 12=2 So, the required interval is x 1 ks As ñ= 68, s=10, k=2. So, the interval is că-ks , ætes) 31 68-10X2, 68t0X2) = (48,88)
Here we first define the chebyshevs theorem so that we can use this theorem correctly then calculate the required term as