Chapter 1.4, Problem 111E

(a)

To determine

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

Answer to Problem 111E

Solution: The calculated values are $\underset{\_}{\left(8489,20,919\right)}$, $\underset{\_}{\left(2274,27,134\right)}$, and $\underset{\_}{\left(-3941,33,349\right)}$.

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,

$\begin{array}{c}\left(\mu \pm \sigma \right)=\left(\mu -\sigma ,\mu +\sigma \right)\\ =\left(14,704-6215,14,704+6215\right)\\ =\left(8489,20,919\right)\end{array}$

Step 2: For calculating 95% of data,

$\begin{array}{c}\left(\mu \pm 2\sigma \right)=\left(\mu -2\sigma ,\mu +2\sigma \right)\\ =\left(14,704-\left(2\times 6215\right),14,704+\left(2\times 6215\right)\right)\\ =\left(2274,27,134\right)\end{array}$

Step 3: For calculating 99.7% of data,

$\begin{array}{c}\left(\mu \pm 3\sigma \right)=\left(\mu -3\sigma ,\mu +3\sigma \right)\\ =\left(14,704-\left(3\times 6215\right),14,704+\left(3\times 6215\right)\right)\\ =\left(-3941,33,349\right)\end{array}$

To determine

To explain: Whether the above rule is applicable or not.

Answer to Problem 111E

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. The normal distribution cannot be used in the provided case. This rule is applicable only in the case of normal distribution.

To determine

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

Answer to Problem 111E

Solution: The calculated values are $\underset{\_}{\left(7158,22,886\right)}$, $\underset{\_}{\left(-706,\text{30,750}\right)}$, and $\underset{\_}{\left(-8570,38,614\right)}$.

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 scores for the students can be obtained by the following steps shown below:

Step 1: For calculating 68% of data,

$\begin{array}{c}\left(\mu \pm \sigma \right)=\left(\mu -\sigma ,\mu +\sigma \right)\\ =\left(15,022-7864,15,022+7864\right)\\ =\left(7158,22,886\right)\end{array}$

Step 2: For calculating 95% of data,

$\begin{array}{c}\left(\mu \pm 2\sigma \right)=\left(\mu -2\sigma ,\mu +2\sigma \right)\\ =\left(15,022-\left(2\times 7864\right),15,022+\left(2\times 7864\right)\right)\\ =\left(-706,\text{30,750}\right)\end{array}$

Step 3: For calculating 99.7% of data,

$\begin{array}{c}\left(\mu \pm 3\sigma \right)=\left(\mu -3\sigma ,\mu +3\sigma \right)\\ =\left(15,022-\left(3\times 7864\right),15,022+\left(3\times 7864\right)\right)\\ =\left(-8570,38,614\right)\end{array}$

To determine

To explain: Whether the above rule is applicable or not.

Answer to Problem 111E

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.

To determine

To explain: Whether the data supports conventional wisdom or not.

Answer to Problem 111E

Solution: 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 men is greater than the mean number of words spoken per day by the women, and the standard deviation of women is less than the standard deviation of men. There is a more concentrated distribution of the number of words spoken by women than men.

(b)

To determine

To explain: The conclusion from the observation.

Answer to Problem 111E

Solution: Both men and women from Mexico tend to speak more than those from United states.

Explanation of Solution

The mean for both men and women is higher for the Mexican study as compared to U.S. study. It shows that both men and women from Mexico tend to speak more than those from the United States.