Question

At Community Hospital, the burn center is experimenting with a new plasma compress treatment. A random sample of n₁ = 302 patients with minor burns received the plasma compress treatment. Of these patients, it was found that 250 had no visible scars after treatment. Another random sample of n₂ = 440patients with minor burns received no plasma compress treatment. For this group, it was found that 85 had no visible scars after treatment. Let p₁ be the population proportion of all patients with minor burns receiving the plasma compress treatment who have no visible scars. Let p₂ be the population proportion of all patients with minor burns not receiving the plasma compress treatment who have no visible scars.

(a) Find a 99% confidence interval for p₁ − p₂. (Round your answers to three decimal places.)

lower limit
upper limit

(b) Explain the meaning of the confidence interval found in part (a) in the context of the problem. Does the interval contain numbers that are all positive? all negative? both positive and negative? At the 99% level of confidence, does treatment with plasma compresses seem to make a difference in the proportion of patients with visible scars from minor burns?

Because the interval contains both positive and negative numbers, we can not say that there is a higher proportion of patients with no visible scars among those who received the treatment.We can not make any conclusions using this confidence interval.     Because the interval contains only negative numbers, we can say that there is a higher proportion of patients with no visible scars among those who did not receive the treatment.Because the interval contains only positive numbers, we can say that there is a higher proportion of patients with no visible scars among those who received the treatment.

Answer 1

a)

sample #1   ----->              
first sample size,     n1=   302          
number of successes, sample 1 =     x1=   250          
proportion success of sample 1 , p̂1=   x1/n1=   0.8278          
                  
sample #2   ----->              
second sample size,     n2 =    440          
number of successes, sample 2 =     x2 =    85          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.1932          
                  
difference in sample proportions, p̂1 - p̂2 =     0.8278   -   0.1932   =   0.6346

level of significance, α =   0.01              
Z critical value =   Z α/2 =    2.576   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.0287          
margin of error , E = Z*SE =    2.576   *   0.0287   =   0.0740
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.635   -   0.0740   =   0.561
upper limit = (p̂1 - p̂2) + E =    0.635   +   0.0740   =   0.709

b)

  
Because the interval contains only positive numbers, we can say that there is a higher proportion of patients with no visible scars among those who received the treatment.