Chapter 9.2, Problem 12E

a.

To determine

Check whether the sample size is large enough to use the confidence interval for the given combinations of n and $\widehat{p}$.

Answer to Problem 12E

Yes, the sample size of 50 is large enough for the interval to be appropriate.

Explanation of Solution

Calculation:

The given information is that $n=50\text{ and }\widehat{p}=0.30$.

The interval is appropriate, if $n\widehat{p}\ge 10\text{ and }n\left(1-\widehat{p}\right)\ge 10$.

Verify the conditions:

Substitute, $n=50\text{ and }\widehat{p}=0.30$ in the above conditions.

$\begin{array}{c}n\widehat{p}=50\left(0.3\right)\\ =15\\ \ge 10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=50\left(1-0.3\right)\\ =50\left(0.7\right)\\ =35\\ \ge 10\end{array}$

Thus, both the conditions are satisfied.

The sample size of 50 is large enough for the interval to be appropriate.

b.

To determine

Explain whether the sample size is large enough to use the confidence interval for the given combinations of n and $\widehat{p}$.

Answer to Problem 12E

No, the sample size of 50 is not large enough for the interval to be appropriate.

Explanation of Solution

Calculation:

The given information is that $n=50\text{ and }\widehat{p}=0.05$.

Verify the conditions:

Substitute, $n=50\text{ and }\widehat{p}=0.05$ in the conditions specified in Part (a).

$\begin{array}{c}n\widehat{p}=50\left(0.05\right)\\ =2.5\\ <10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=50\left(1-0.05\right)\\ =50\left(0.95\right)\\ =47.5\\ \ge 10\end{array}$

Thus, the first condition is not satisfied.

The sample size of 50 is not large enough for the interval to be appropriate.

c.

To determine

Define whether the sample size is large enough to use the confidence interval for the given combinations of n and $\widehat{p}$.

Answer to Problem 12E

No, the sample size of 15 is not large enough for the interval to be appropriate.

Explanation of Solution

Calculation:

The given information is that $n=15\text{ and }\widehat{p}=0.45$.

Verify the conditions:

Substitute, $n=15\text{ and }\widehat{p}=0.45$ in the conditions specified in Part (a).

$\begin{array}{c}n\widehat{p}=15\left(0.45\right)\\ =6.75\\ <10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=15\left(1-0.45\right)\\ =50\left(0.55\right)\\ =8.25\\ <10\end{array}$

Thus, the both the conditions are not satisfied.

The sample size of 15 is not large enough for the interval to be appropriate.

d.

To determine

Elucidate whether the sample size is large enough to use the confidence interval for the given combinations of n and $\widehat{p}$.

Answer to Problem 12E

No, the sample size of 100 is not large enough for the interval to be appropriate.

Explanation of Solution

Calculation:

The given information is that $n=100\text{ and }\widehat{p}=0.01$.

Verify the conditions:

Substitute $n=100\text{ and }\widehat{p}=0.01$ in the conditions specified in Part (a).

$\begin{array}{c}n\widehat{p}=100\left(0.01\right)\\ =1\\ <10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=100\left(1-0.01\right)\\ =100\left(0.99\right)\\ =99\\ \ge 10\end{array}$

Thus, the first condition is not satisfied.

The sample size of 100 is not large enough for the interval to be appropriate.

e.

To determine

Examine whether the sample size is large enough to use the confidence interval for the given combinations of n and $\widehat{p}$.

Answer to Problem 12E

Yes, the sample size of 100 is large enough for the interval to be appropriate.

Explanation of Solution

Calculation:

The given information is that $n=100\text{ and }\widehat{p}=0.70$.

Verify the conditions:

Substitute, $n=100\text{ and }\widehat{p}=0.70$ in the conditions specified in Part (a).

$\begin{array}{c}n\widehat{p}=100\left(0.70\right)\\ =70\\ \ge 10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=100\left(1-0.70\right)\\ =100\left(0.30\right)\\ =30\\ \ge 10\end{array}$

Thus, both the conditions are satisfied.

The sample size of 100 is large enough for the interval to be appropriate.

f.

To determine

Describe whether the sample size is large enough to use the confidence interval for the given combinations of n and $\widehat{p}$.

Answer to Problem 12E

Yes, the sample size of 40 is large enough for the interval to be appropriate.

Explanation of Solution

Calculation:

The given information is that $n=40\text{ and }\widehat{p}=0.25$.

Verify the conditions:

Substitute, $n=40\text{ and }\widehat{p}=0.25$ in the conditions specified in Part (a).

$\begin{array}{c}n\widehat{p}=40\left(0.25\right)\\ =10\\ \ge 10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=40\left(1-0.25\right)\\ =40\left(0.75\right)\\ =30\\ \ge 10\end{array}$

Thus, both the conditions are satisfied.

The sample size of 40 is large enough for the interval to be appropriate.

g.

To determine

Justify whether the sample size is large enough to use the confidence interval for the given combinations of n and $\widehat{p}$.

Answer to Problem 12E

Yes, the sample size of 60 is large enough for the interval to be appropriate.

Explanation of Solution

Calculation:

The given information is that $n=60\text{ and }\widehat{p}=0.25$.

Verify the conditions:

Substitute, $n=60\text{ and }\widehat{p}=0.25$ in the conditions specifies in Part (a).

$\begin{array}{c}n\widehat{p}=60\left(0.25\right)\\ =15\\ \ge 10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=60\left(1-0.25\right)\\ =60\left(0.75\right)\\ =45\\ \ge 10\end{array}$

Thus, both the conditions are satisfied.

The sample size of 60 is large enough for the interval to be appropriate.

h.

To determine

Check whether the sample size is large enough to use the confidence interval for the given combinations of n and $\widehat{p}$.

Answer to Problem 12E

No, the sample size of 80 is not large enough for the interval to be appropriate.

Explanation of Solution

Calculation:

The given information is that $n=80\text{ and }\widehat{p}=0.1$.

Verify the conditions:

Substitute, $n=80\text{ and }\widehat{p}=0.1$ in the conditions specifies in Part (a).

$\begin{array}{c}n\widehat{p}=80\left(0.1\right)\\ =8\\ <10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=80\left(1-0.1\right)\\ =80\left(0.90\right)\\ =72\\ \ge 10\end{array}$

Thus, the first condition is not satisfied.

The sample size of 80 is not large enough for the interval to be appropriate.