Chapter 9.2, Problem 31E

The article “Hospitals Dispute Medtronic Data on Wires” (The Wall Street Journal, February 4, 2010) describes several studies of the failure rate of defibrillators used in the treatment of heart problems. In one study conducted by the Mayo Clinic, it was reported that failures were experienced within the first 2 years by 18 of 89 patients under 50 years old and 13 of 362 patients age 50 and older who received a particular type of defibrillator. Assume it is reasonable to regard these two samples as representative of patients in the two age groups who receive this type of defibrillator.

1. a. Construct and interpret a 95% confidence interval for the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator.
2. b. Construct and interpret a 99% confidence interval for the proportion of patients age 50 and older who experience a failure within the first 2 years after receiving this type of defibrillator.
3. c. Suppose that the researchers wanted to estimate the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator to within 0.03 with 95% confidence. How large a sample should be used? Use the results of the study as a preliminary estimate of the population proportion. (Hint: See Example 9.6.)

To determine

Find the 95% confidence interval for the proportion of patients under 50 years’ old who experience a failure within the first 2 years after receiving a particular type of defibrillator and interpret it.

Answer to Problem 31E

The 95% confidence interval for the proportion of patients under 50 years’ old who experience a failure within the first 2 years after receiving a particular type of defibrillator is (0.119,0.286).

Interpretation of confidence interval is “One can be 95% confident that the proportion of patients under 50 years’ old who experience a failure within the first 2 years after receiving a particular type of defibrillator is between 0.119 and 0.286”.

Explanation of Solution

Calculation:

It was found that failures were experienced within the first two years by 18 of the 89 patients under 50 years old and 13 of 362 patients at the age of 50 or older. The assumption is that the sample is a representative of the population of the patients in two age groups who received he treatment.

The confidence interval for a population proportion p is appropriate when,

• The sample size is large. That is, when $n\widehat{p}\ge 10\text{ and }n\left(1-\widehat{p}\right)\ge 10$,
• The sample size is small relative to the population size. That is, the sample size should not be more than 10% of the population size.
• The sampling method is simple random sampling.

Then, the general formula for finding the confidence interval for a population proportion p is, $\widehat{p}\pm \left(z\text{ critical value}\right)\sqrt{\frac{\widehat{p}\left(1-\widehat{p}\right)}{n}}$, where $\widehat{p}$ is the sample proportion and n is the sample size.

The number of patients under 50 years’ old who experience a failure within the first 2 years after receiving a particular type of defibrillator is 18 of the 89 patients. That is, the sample proportion is, $\frac{18}{89}$ .

Check conditions:

Substitute $n=89\text{ and }\widehat{p}=\frac{18}{89}$. Then,

$\begin{array}{c}n\widehat{p}=89\left(\frac{18}{89}\right)\\ =18\\ >10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=89\left(1-\frac{18}{89}\right)\\ =89\left(\frac{71}{89}\right)\\ =71\\ >10\end{array}$

Since, both the conditions are satisfied and the sample is large enough which makes the confidence interval appropriate.

The population is considered the patients under 50 years’ old who experience a failure within the first 2 years after receiving a particular type of defibrillator. A sample of 89 patients is selected for the survey. Since the population considered is large, the sample size will be smaller than 10% of the population size.

It is assumed that the sample is a representative of the population of the patients in two age groups who received he treatment. Thus, it is reasonable to consider the sample as a random sample from the population.

The 95% confidence interval for proportion is calculated below:

Software procedure:

Step-by-step procedure to obtain the confidence interval using MINITAB software is given below:

• Choose Stat > Basic statistics>1-Sample proportion.
• In Summarized data, enter Number of events as 18 and Number of trials as 89.
• Check Options, enter Confidence level as 95.
• Choose not equal in alternative hypothesis.
• In method select Normal Approximation.
• Click OK in all dialogue boxes.

The output using the MINITAB software is given below:

From the MINITAB output, the confidence interval is (0.119,0.286).

Thus, the 95% confidence interval for the proportion of patients under 50 years’ old who experience a failure within the first 2 years after receiving a particular type of defibrillator is (0.119,0.286).

Interpretation of confidence interval is “One can be 95% confident that the proportion of patients under 50 years’ old who experience a failure within the first 2 years after receiving a particular type of defibrillator is between 0.119 and 0.286”.

To determine

Find the 99% confidence interval for the proportion of patients age 50 years and old who experience a failure within the first 2 years after receiving a particular type of defibrillator and interpret it.

Answer to Problem 31E

The 99% confidence interval for the proportion of patients age 50 years and old who experience a failure within the first 2 years after receiving a particular type of defibrillator is (0.011,0.061).

Interpretation of confidence interval is “One can be 99% confident that the proportion of patients age 50 years and old who experience a failure within the first 2 years after receiving a particular type of defibrillator is between 0.011 and 0.061”.

Explanation of Solution

Calculation:

The number of patients age 50 years and old who experience a failure within the first 2 years after receiving a particular type of defibrillator is 13 of the 362 patients. That is, the sample proportion is, $\frac{13}{362}$ .

Check conditions:

Substitute $n=362\text{ and }\widehat{p}=\frac{13}{362}$. Then,

$\begin{array}{c}n\widehat{p}=362\left(\frac{13}{362}\right)\\ =13\\ >10\end{array}$

$\begin{array}{c}n\left(1-\widehat{p}\right)=362\left(1-\frac{13}{362}\right)\\ =362\left(\frac{349}{362}\right)\\ =349\\ >10\end{array}$

Since, both the conditions are satisfied the sample is large enough which makes the confidence interval appropriate.

The population is considered the patients age 50 years and old who experience a failure within the first 2 years after receiving a particular type of defibrillator. A sample of 362 patients is selected for the survey. Since the population considered is large, the sample size will be smaller than 10% of the population size.

It is assumed that the sample is a representative of the population of the patients in two age groups who received a particular type of defibrillator. Thus, it is reasonable to consider the sample as a random sample from the population.

The 95% confidence interval for proportion is calculated below:

Software procedure:

Step-by-step procedure to obtain the confidence interval using MINITAB software is given below:

• Choose Stat > Basic statistics>1-Sample proportion.
• In Summarized data, enter Number of events as 18 and Number of trials as 89.
• Check Options, enter Confidence level as 95.
• Choose not equal in alternative hypothesis.
• In method select Normal Approximation.
• Click OK in all dialogue boxes.

The output using the MINITAB software is given below:

From the MINITAB output, the confidence interval is (0.011,0.061).

Thus, the 99% confidence interval for the proportion of patients age 50 years and old who experience a failure within the first 2 years after receiving a particular type of defibrillator is (0.011,0.061).

Interpretation of confidence interval is “One can be 99% confident that the proportion of patients age 50 years and old who experience a failure within the first 2 years after receiving a particular type of defibrillator is between 0.011 and 0.061”.

To determine

Identify the sample size required to estimate the proportion of patients under the age 50 within 0.03 and a 95% confidence.

Answer to Problem 31E

The sample size needed to estimate the proportion of patients under the age 50 within 0.03 and a 95% confidence is 689.

Explanation of Solution

Calculation:

The preliminary estimate of p obtained from the study is $\frac{18}{89}$.

The sample size required to estimate the population proportion within an amount M with a 95% level of confidence is, $p\left(1-p\right){\left(\frac{1.96}{M}\right)}^2$.

Substitute, $p=\frac{18}{89},M=0.03$. Then, the number of students to be included in the sample to estimate the proportion within 0.03 with 95% confidence is,

$\begin{array}{c}\text{Sample size = }p\left(1-p\right){\left(\frac{1.96}{M}\right)}^2\\ =\left(\frac{18}{89}\right)\left(1-\frac{18}{89}\right){\left(\frac{1.96}{0.03}\right)}^2\\ =\left(0.2022\right)\left(0.7977\right){\left(65.33\right)}^2\\ =0.1613\left(4,268.44\right)\\ =688.49\\ \approx 689\end{array}$

Thus, the sample size needed to estimate the proportion of patients under the age 50 within 0.03 and a 95% confidence is 689.