Chapter 1.4, Problem 110E

(a)

To determine

To find: The distribution using $68-95-99.7$ rule.

Answer to Problem 110E

Solution: The calculated values are $\underset{\_}{\left(7856,20,738\right)}$, $\underset{\_}{\left(1415,27,179\right)}$, and $\underset{\_}{\left(-5026,33,620\right)}$.

Explanation of Solution

Calculation: The $68-95-99.7$ rule states that in the normal distribution, 68% of data falls under $\left(\mu \pm \sigma \right)$, 95% of the data falls under $\left(\mu \pm 2\sigma \right)$, and 99.7% of the data falls under $\left(\mu \pm 3\sigma \right)$.

With the given values of mean and standard deviation, the range of the number of words can be obtained by the following steps:

Step 1: For calculating 68% of data,

$\begin{array}{c}\mu \pm \sigma =\left(\mu -\sigma ,\mu +\sigma \right)\\ =\left(14,297-6441,14,297+6441\right)\\ =\left(7856,20,738\right)\end{array}$

Step 2: For calculating 95% of data,

$\begin{array}{c}\mu \pm 2\sigma =\left(\mu -2\sigma ,\mu +2\sigma \right)\\ =\left(14,297-\left(2\times 6441\right),14,297+\left(2\times 6441\right)\right)\\ =\left(1415,27,179\right)\end{array}$

Step 3: For calculating 99.7% of data,

$\begin{array}{c}\mu \pm 3\sigma =\left(\mu -3\sigma ,\mu +3\sigma \right)\\ =\left(14,297-\left(3\times 6441\right),14,297+\left(3\times 6441\right)\right)\\ =\left(-5026,33,620\right)\end{array}$

(b)

To determine

Whether the above rule is applicable or not.

Answer to Problem 110E

Solution: This rule is not applicable.

Explanation of Solution

From the above calculation, it is clear that this rule is not applicable here because it is giving negative values and words spoken cannot be negative. Therefore, the normal distribution cannot be used in this case. This rule is applicable only when the data has a normal distribution.

(c)

To determine

To find: The distribution using $68-95-99.7$ rule.

Answer to Problem 110E

Solution: The calculated values are obtained as $\underset{\_}{\left(5004,23,116\right)}$, $\underset{\_}{\left(-4052,\text{32,172}\right)}$, and $\underset{\_}{\left(-13,108,41,228\right)}$.

Explanation of Solution

Calculation: The $68-95-99.7$ rule states that in the normal distribution, 68% of data falls under $\left(\mu \pm \sigma \right)$, 95% of the data falls under $\left(\mu \pm 2\sigma \right)$, and 99.7% of the data falls under $\left(\mu \pm 3\sigma \right)$.

With the given values of mean and standard deviation, the range of the number of words can be obtained by the following steps:

Step 1: For calculating 68% of data,

$\begin{array}{c}\mu \pm \sigma =\left(\mu -\sigma ,\mu +\sigma \right)\\ =\left(14,060-9056,14,060+9056\right)\\ =\left(5004,23,116\right)\end{array}$

Step 2: For calculating 95% of data,

$\begin{array}{c}\mu \pm 2\sigma =\left(\mu -2\sigma ,\mu +2\sigma \right)\\ =\left(14,060-\left(2\times 9056\right),14,060+\left(2\times 9056\right)\right)\\ =\left(-4052,\text{32,172}\right)\end{array}$

Step 3: For calculating 99.7% of data,

$\begin{array}{c}\mu \pm 3\sigma =\left(\mu -3\sigma ,\mu +3\sigma \right)\\ =\left(14,060-\left(3\times 9056\right),14,060+\left(3\times 9056\right)\right)\\ =\left(-13,108,41,228\right)\end{array}$

To determine

Whether the above rule is applicable or not.

Answer to Problem 110E

Solution: This rule is also not applicable.

Explanation of Solution

From the above calculation, it is clear that this rule is not applicable here because it is also providing negative values, which cannot be possible.

(d)

To determine

Whether the data supports conventional wisdom or not.

Answer to Problem 110E

Solution: The data does not support conventional wisdom.

Explanation of Solution

It can be seen from the question that the mean number of words spoken per day by the women is greater than the mean number of words spoken per day by the men and the standard deviation of women is less than the standard deviation of men. There is a more concentrated distribution of number of words spoken for women than men.