Question

An article 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 12 of 97 patients under 50 years old and 13 of 363 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.

(a) Construct 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. (Round your answers to three decimal places.)
(  ,  )

Interpret the interval.

We are 95% confident that the proportion of all patients who experience a failure after receiving this type of defibrillator is within this interval.We are confident that the proportion of all patients under 50 years old who experience a failure within the first two years after receiving this type of defibrillator is within this interval at least 95% of the time.    We are confident that 95% of all patients under 50 years old who experience a failure within the first two years after receiving this type of defibrillator are within this interval.We are 95% confident that the proportion of all patients under 50 years old who experience a failure within the first two years after receiving this type of defibrillator is within this interval.

(b) Construct 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. (Round your answers to three decimal places.)
(  ,  )

Interpret the interval.

We are 99% confident that the proportion of all patients who experience a failure after receiving this type of defibrillator is within this interval.We are 99% confident that the proportion of all patients age 50 or older who experience a failure within the first two years after receiving this type of defibrillator is within this interval.    We are confident that the proportion of all patients age 50 or older who experience a failure within the first two years after receiving this type of defibrillator is within this interval at least 99% of the time.We are confident that 99% of all patients age 50 or older who experience a failure within the first two years after receiving this type of defibrillator is within this interval.

(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. (Enter your answer as a whole number.)
patients

Answer 1

(a) The calculation of the sample proportion is as shown below: = 0.124 The calculation of the Z score corresponding to 95% confidence interval in Excel is as shown below: fc =NORMSINV(1-0.05/2) D E F 1.96 The calculation of the confidence interval is as shown below: CI= 8+2.$(1-) = 0.124£1.96, 0.124(1–0.124) 97 = (0.058.0.190) The confidence interval is (0.058, 0.190) Interpret the interval: We are 95% confident that the proportion of all patients under 50 years old who experience a failure within the first two years after receiving this type of defibrillator is within this interval. (b) The calculation of the sample proportion is as shown below: 13 P 363 = 0.036

The calculation of the Z score corresponding to 99% confidence interval in Excel is as shown below: for ENORMSINV(1-0.01/2) F L2.58 The calculation of the confidence interval is as shown below: CI=PLV n CI= PIZ, Þ(1-6) = 0.03612.58 10.036(1-0.036) 363 =(0.011.0.061) The confidence interval is (0.011, 0.061) Interpret the interval: We are confident that 99% of all patients age 50 or older who experience a failure within the first two years after receiving this type of defibrillator is within this interval. (c) The calculation of the sample size is as shown below: L2xp(1-0) n = ME 1.962x0.124x(1-0.124) 0.032 = 463.66 = 464 Therefore, a sample of size 464 should be used.

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