Question

a quantitative data set has mean 24 and standard deviation 3.
a. what percentage of the observations lie between 15 and 33?
at least
%

b. if the data set contains 78 pieces of data, how many data values are promised between 15 and 33?
at least
data values.

Explanation:

Step1: Calculate the number of standard - deviations away from the mean

First, find how many standard deviations 15 and 33 are from the mean $\mu = 24$ with standard deviation $\sigma=3$.
For $x = 15$, $z_1=\frac{15 - 24}{3}=\frac{- 9}{3}=-3$.
For $x = 33$, $z_2=\frac{33 - 24}{3}=\frac{9}{3}=3$.

Step2: Apply Chebyshev's theorem

Chebyshev's theorem states that for any number $k>0$, the proportion of the data that lies within $k$ standard deviations of the mean is at least $1-\frac{1}{k^{2}}$. Here $k = 3$.
So the proportion of data within 3 standard deviations of the mean is at least $1-\frac{1}{3^{2}}=1-\frac{1}{9}=\frac{8}{9}\approx0.8889$ or $88.89\%$.

Step3: Calculate the number of data values

If the data - set contains $n = 78$ pieces of data, and the proportion of data within 3 standard deviations of the mean is at least $\frac{8}{9}$, then the number of data values within this range is at least $n\times(1 - \frac{1}{k^{2}})$.
$78\times\frac{8}{9}=\frac{624}{9}\approx69.33$. Since the number of data values is an integer, the number of data values is at least 70.

Answer:

a. 88.89
b. 70