Question

Assume the normal distribution of data has a mean of 14 and a standard Deviation of 3. use the 65-95-99.7 rule to find the percentage of values that lie below 8

Answer 1

By the 65-95-99.7 rule,

`\begin{gathered} 65\text{ \% of the distribution lies below }\bar{x}+\sigma\text{ and above }\bar{x}-\sigma \\ 95\text{ \% of the distribution lies below }\bar{x}+2\sigma\text{ and above }\bar{x}-2\sigma \\ 99.7\text{ \% of the distribution lies below }\bar{x}+3\sigma\text{ and above }\bar{x}-3\sigma \end{gathered}`

By symmetry,

`\begin{gathered} 47.5\text{ \% of the distribution lies above }\bar{x}-\sigma\text{ and below }\bar{x} \\ \text{ Hence,} \\ 2.5\text{ \% of the values lies below }\bar{x}-\sigma \end{gathered}`

In our case,

`\bar{x}=14,\sigma=3`

Therefore,

`\begin{gathered} 8=14-6=14-2(3) \\ \text{Hence,'} \\ 8=\bar{x}-2\sigma \end{gathered}`

Hence, 2.5 % of the values lie below 8