Chapter PV, Problem 4RE

(a)

To determine

To explain would you always get $507$ votes for Bush and $483$ for Kerry.

Answer to Problem 4RE

No, we would not always get $507$ votes for Bush and $483$ for Kerry.

Explanation of Solution

It is given in the question that the official result of the election showed that Bush received $50.7\%$ of the vote and John Kerry received $48.3\%$ . Nader running a third party candidate picked up only $0.4\%$ vote. Thus, no, we would not always get $507$ votes for Bush and $483$ for Kerry because it is rare that you obtain the exact same sample results for different samples.

(b)

To determine

To explain your sample proportion for voters for Bush should be between what two numbers.

Answer to Problem 4RE

The sample proportion for voters for Bush should be between $0.4754\text{ and 0}\text{.5386}$ .

Explanation of Solution

It is given in the question that the official result of the election showed that Bush received $50.7\%$ of the vote and John Kerry received $48.3\%$ . Nader running a third party candidate picked up only $0.4\%$ vote. As it is given that,

  $\begin{array}{l}p=50.7\%=0.507\\ n=1000\\ c=0.95=95\%\end{array}$

The mean of the sampling distribution of the sample proportion is p , while the standard deviation is $\sqrt{\frac{p(1-p)}{n}}$ .

Thus, the $68-95-99.7$ rule tells us that $95\%$ of the sample proportions is within two standard deviations of the mean. So, we have,

  $\begin{array}{l}p-2\sqrt{\frac{p(1-p)}{n}}\\ =0.507-2\sqrt{\frac{0.507(1-0.507)}{1000}}\\ =0.4754\\ p+2\sqrt{\frac{p(1-p)}{n}}\\ =0.507+2\sqrt{\frac{0.507(1-0.507)}{1000}}\\ =0.5386\end{array}$

Thus, the sample proportion for voters for Bush should be between $0.4754\text{ and 0}\text{.5386}$ .

(c)

To determine

To explain your sample proportion for voters for Nader should be between what two numbers.

Answer to Problem 4RE

The sample proportion for voters for Nader cannot be calculated.

Explanation of Solution

It is given in the question that the official result of the election showed that Bush received $50.7\%$ of the vote and John Kerry received $48.3\%$ . Nader running a third party candidate picked up only $0.4\%$ vote. As it is given that,

  $\begin{array}{l}p=0.4\%=0.004\\ n=1000\\ c=0.95=95\%\end{array}$

The mean of the sampling distribution of the sample proportion is p , while the standard deviation is $\sqrt{\frac{p(1-p)}{n}}$ . Since it will be difficult to collect a random sample of the population as some might not be willing to participate in the study and the condition of success and failure is not satisfied in this as it is not at least ten because the votes given to Nader is very less to satisfy the condition.

(d)

To determine

To explain why would you expect the sample proportion of Nader votes to vary more, less or about the same as the sample proportion of Bush votes.

Answer to Problem 4RE

We would expect the sample proportion of Nader votes to vary less as the sample proportion of Bush votes.

Explanation of Solution

It is given in the question that the official result of the election showed that Bush received $50.7\%$ of the vote and John Kerry received $48.3\%$ . Nader running a third party candidate picked up only $0.4\%$ vote. Thus, we note that we would expect the sample proportion of Nader votes to vary less as the sample proportion of Bush votes because those people who are willing to participate in the study are more likely to be supporters of Bush than the general population which causes less variation.