**Question 19** The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 58 ounces and a standard deviation of 6 ounces. Use the Empirical Rule. Suggestion: sketch the distribution in order to answer these questions. a) 95% of the widget weights lie between _______ and _______. b) What percentage of the widget weights lie between 52 and 70 ounces? ______% c) What percentage of the widget weights lie above 40? ______% [ ] Calculator [Submit Question Button] --- Below Question 19, there is a start of Question 20, with a partially visible table: - **Probability Scores** - 0.3 | 1 - 0.05 | 6 - 0.25 | 8 (Note: The table appears to display probabilities associated with different scores, but is cut off.)

Given $\mu =58 and \sigma =6$.

a) The empirical rule states that 95% of data values fall within two standard deviations of mean, i.e.
$\begin{array}{rcl}\mu \pm 2\sigma & =& 58\pm 2\left(6\right)\\ & =& 58\pm 12\\ & =& \left(46 , 70\right)\end{array}$

Therefore, 46 and 70 is the answer.

b)

Approximately 68% of the data values fall within one standard deviation of mean, i.e.
$\begin{array}{rcl}\mu \pm \sigma & =& 58\pm 6\\ & =& \left(52 , 64\right)\end{array}$

Approximately 99.7% of the data values fall within three standard deviations of mean, i.e.
$\begin{array}{rcl}\mu \pm 3\sigma & =& 58\pm 3\left(6\right)\\ & =& 58\pm 18\\ & =& \left(40 , 76\right)\end{array}$

Again by empirical rule,
$\left(\frac{68}{2}\right)\%=34\% data lies within \left(\mu -\sigma ,\mu \right) and the rest of 34\% data lies within \left(\mu ,\mu +\sigma \right)$.

Similarly,
$\left(\frac{95}{2}\right)\%=47.5\% data lies within \left(\mu -2\sigma ,\mu \right) and the rest of 47.5\% data lies within \left(\mu ,\mu +2\sigma \right)$.

Then, the required percentage is
$\begin{array}{rcl}required percentage& =& Percentage which falls within \left(\mu ,\mu +\sigma \right)-Percentage which falls within \left(\mu -\sigma ,\mu \right)\\ & =& 47.5-34\\ & =& 13.5\%\end{array}$

Therefore, the answer is 13.5%.