Mathematics: A quantitative data set of size 80 has mean 30 and standard deviation 5. At least how many observations lie between 15 and 45?

A quantitative data set of size 80 has mean 30

Ответ:

Chewychipsx4023

18.01.2022

At least 71 observations must lie inside the interval (15, 45).

Step-by-step explanation:

According with the Chebyshev’s Theorem, for any data set, the proportion (or percentage) of values that fall within k standard deviations from mean (that is, in the interval $(\bar{x}-ks,\bar{x}+ks)$ ) is at least $1-\frac{1}{k^2}$ , where k > 1.

Given:

$\bar{x}=30\\s=5$

The interval (15, 45) is the one that is formed by adding and subtracting three standard deviations from the mean.

$(\bar{x}-ks,\bar{x}+ks)$

$(30-3\cdot 5,30+3\cdot 5)=(15,45)$

By Chebyshev’s Theorem, at least $1-\frac{1}{3^2}=\frac{8}{9}$ of the data are within this interval. Since $\frac{8}{9}$ of 80 is 71.11, this means that at least 71.11 observations are in the interval. But one cannot take a fractional observation, so we conclude that at least 71 observations must lie inside the interval (15, 45).