Q. 6.105 - Elementary Statistics

Question

In 1905, R. Pearl published the article "Biometrical Studies on Man. 1. Variation and Correlation in Brain Weight". According to the study, brain weights of Swedish men are normally distributed with mean $1.4kg$ and standard deviation $0.11kg$. Apply the empirical rule to fill in the blanks.

Part (a): Approximately $68\%$ of Swedish men have brain weights between ___ and ___.

Part (b): Approximately $95\%$of Swedish men have brain weights between ___ and ___.

Part (c): Approximately $99.7\%$ of Swedish men have brain weights between ___ and ___.

Part (d): Draw graphs similar to those in Fig. $6.27$on page $272$ to portray your results.

Step-by-Step Solution

Answer

Part (a): Approximately $68\%$of Swedish men have brain weights between $1.29kg$ and$1.51kg$.

Part (b): Approximately $95\%$ of Swedish men have brain weights between $1.18kg$ and $1.62kg$.

Part (c): Approximately $99.7\%$ of Swedish men have brain weights between $1.07kg$ and $1.73kg$.

Part (d): Graph of approximately $68\%$ of Swedish men is given below,

Graph of approximately $95\%$ of Swedish men is given below,

Graph of approximately $99.7\%$ of Swedish men is given below,

Part (a) Step 1. Given information.

Consider the given question,

The brain weights are approximately normal with mean $1.4kg$ and standard deviation $0.11kg$.

Part (a) Step 2. Determine the approximate weight within 68 % .

By property of empirical rule, approximately $68\%$ of values lie in the interval,

$$\begin{gathered} \left(\mu -\sigma ,\mu +\sigma \right)=\left(1.4-0.11,1.4+0.11\right) \\ =\left(1.29,1.51\right) \end{gathered}$$

Thus, approximately $68\%$of all Swedish men have brain weights between $1.29,1.51kg$.

Part (b) Step 1. Determine the approximate weight within 95 % .

By property of empirical rule, approximately $95\%$ of values lie in the interval,

$$\begin{gathered} \left(\mu -2\sigma ,\mu +2\sigma \right)=\left(1.4-\left(2\right)0.11,1.4+\left(2\right)0.11\right) \\ =\left(1.18,1.62\right) \end{gathered}$$

Thus, approximately $95\%$ of all Swedish men have brain weights between $1.18,1.62kg$.

Part (c) Step 1. Determine the approximate weight within 99 . 7 % .

By property of empirical rule, approximately $99.7\%$ of values lie in the interval,

$$\begin{gathered} \left(\mu -3\sigma ,\mu +3\sigma \right)=\left(1.4-\left(3\right)0.11,1.4+\left(3\right)0.11\right) \\ =\left(1.07,1.73\right) \end{gathered}$$

Thus, approximately $99.7\%$ of all Swedish men have brain weights between $1.07,1.73kg$.

Part (d) Step 1. Drawing graph representing approximately 68 % , 95 % , 99 . 7 % of Swedish men.

On drawing graph representing approximately $68\%$of Swedish men,

On drawing graph representing approximately $95\%$of Swedish men,

On drawing graph representing approximately $99.7\%$ of Swedish men,