Question

In a study of self-adhesive bandages, a doctor wanted to see if there was truly a difference in the healing time between the selfadhesive bandages preloaded with antibiotic and the self-adhesive bandages used with an antibiotic applied to the wound. The doctor randomly selected 100 patients to participate in the study and divided them into two groups of 50 -Group 1 using the bandages preloaded with antibiotic and Group 2 using the bandages with the same amount of antibiotic applied to the wound first. In both groups, the amount of time, in days, it took for the wound to heal was recorded. In Group 1, the mean healing time was 1.60 days with a standard deviation of 0.51, and the mean for Group 2 was 3.28 days with a standard deviation of 0.76. (a) Which of the following is the best way to describe this study? Experiment with 2 treatments: the bandages with antibiotic and the bandages with antibiotic applied first Observational study Experiment with 1 treatment: the type of antibiotic used Experiment with 1 treatment: the types of bandages used (b) Calculate the critical value we would use for a 99% confidence interval for difference in the population means? (Use a table or technology. Round your answer to three decimal places.) x (c) Find the 99% confidence interval for the difference in true mean healing time in wounds between the bandages with preloaded antibiotic and the bandages with antibiotic applied to the wound first. (Round your answers to two decimal places.) We can be 99% confident that the difference in true mean healing time in wounds for the bandages with preloaded antibiotic and the bandages with antibiotic applied to the wound first is between A confidence interval is of the form Point Estimate ± Margin of Error., A confidence interval is of the form Point Estimate ± Margin of Error.). (d) Determine whether the following statement is true or false. If zero lies in the interval, then there is sufficient evidence of a significant difference between the mean healing times for the 2 different bandages. True False

Answer 1

Step-by-step explanation:

(a) The best way to describe this study is: Experiment with 2 treatments: the bandages with antibiotic and the bandages with antibiotic applied first.

(b) To calculate the critical value for a 99% confidence interval for the difference in population means, we need to find the t-value. Since the sample size is relatively large (n = 50), we can use the normal distribution approximation. The critical value for a 99% confidence level corresponds to the z-value at the 0.995 percentile. Using a standard normal distribution table or technology, the critical value is approximately 2.576 (rounded to three decimal places).

(c) To calculate the 99% confidence interval for the difference in true mean healing time between the two types of bandages, we can use the formula:

Confidence Interval = (mean1 - mean2) ± (critical value * standard error)

where:

mean1 = mean healing time for Group 1 (bandages with preloaded antibiotic)

mean2 = mean healing time for Group 2 (bandages with antibiotic applied first)

critical value = 2.576 (from part b)

standard error = sqrt[(variance1/n1) + (variance2/n2)]

Substituting the given values into the formula:

mean1 = 1.60

mean2 = 3.28

standard deviation1 = 0.51

standard deviation2 = 0.76

n1 = n2 = 50

standard error = sqrt[(0.51^2/50) + (0.76^2/50)]

≈ sqrt[0.002601 + 0.009216]

≈ sqrt(0.011817)

≈ 0.1086 (rounded to four decimal places)

Confidence Interval = (1.60 - 3.28) ± (2.576 * 0.1086)

= -1.68 ± 0.2798

= (-1.96, -1.40) (rounded to two decimal places)

Therefore, we can be 99% confident that the difference in true mean healing time in wounds for the bandages with preloaded antibiotic and the bandages with antibiotic applied to the wound first is between -1.96 and -1.40 days.

(d) The statement is true. If the confidence interval includes zero, it means that zero is within the range of plausible values for the difference in mean healing times. This suggests that there is insufficient evidence to conclude that there is a significant difference between the mean healing times for the two different bandages.