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.

Consider the given question,

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

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

localid="1652509206052" $$\begin{gathered} \left(\mu -\sigma ,\mu +\sigma \right)=\left(1.4-0.11,1.4+0.11\right) \\ =\left(1.29,1.51\right) \end{gathered}$$

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

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 betweenrole="math" localid="1652509466598" $1.18,1.62kg$.

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 betweenrole="math" localid="1652509462024" $1.07,1.73kg$.

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,