Chapter 2, Problem 6CRE

(a)

To determine

To find: the range of lengths of pregnant horse

Answer to Problem 6CRE

Approximately $99.7\%$ horse pregnancies fall between 327 to 345 days.

Explanation of Solution

Given:

Mean=336 days

Standard deviation= 3 days.

Calculation:

Bigger animals tend to carry their young longer before birth. The length of horse pregnanciesfrom conception to birth varies according to a roughly Normal distribution with mean 336 days and standard deviation 3 days.

That is, $\mu =336$ and $\sigma =3$

Using the $68-95-99.7$ rule, almost $99.7\%$ horse pregnancies fall within $3\sigma$ of $\mu$

  $\begin{array}{l}\mu \pm 3\sigma =336\pm 3(3)\\ =336\pm 9\\ =336-9,336+9\\ =(327,345)\end{array}$

Approximately $99.7\%$ horse pregnancies fall between 327 to 345 days.

Conclusion:

Therefore, $99.7\%$ horse pregnancies fall between 327 to 345 days.

(b)

To determine

To find:the percent of horse pregnancies which is longer than 339 days.

Answer to Problem 6CRE

  

Explanation of Solution

Calculation:

Using the $68-95-99.7$ rule, almost $68\%$ horse pregnancies fall within $\sigma$ of the mean $\mu$

  $\begin{array}{l}\mu \pm \sigma =336\pm 3\\ =336\pm 3\\ =336-3,336+3\\ =(333,339)\end{array}$

The result is shown in the below graph.

  

Since $16\%$ is one standard deviation above the mean, so according to $68-95-99.7$ rule $116\%$ ofthe horse pregnancies last longer than 339 days. This is because $68\%$ are within one standarddeviation of the mean, so $32\%$ are more than one standard deviation from the mean and half ofthose are greater than 339.

Conclusion:

Therefore, $68\%$ horse pregnancies are longer than 339 days.