Margin of Error + Confidence level

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dr. charles kennedy (expertsolution)answer:to find the confidence interval for the difference in means μd = μ1 - μ2 using the paired data, we first need to calculate the differences d = x1 - x2 for each pair of observations. we can then find the sample mean and standard deviation of these differences, and use them to construct the confidence interval using the t-distribution.
  
calculating the differences and their statistics:
  
             case
            1
            2
            3
            4
            5
                 treatment 1
         23
         27
         32
         26
         27
             treatment 2
         19
         29
         26
         20
         21
             difference
         4
         -2
         6
         6
         6

the sample mean of the differences is:
  
d̄ = (4 - 2 + 6 + 6 + 6) / 5 = 4
  
the sample standard deviation of the differences is:
  
s = √[((4 - 4)² + (-2 - 4)² + (6 - 4)² + (6 - 4)² + (6 - 4)²) / (5 - 1)] = √[(0 + 36 + 4 + 4 + 4) / 4] = √(12) ≈ 3.46
  
using a t-distribution with n - 1 = 4 degrees of freedom and a 99% confidence level, we can find the critical value t*:
  
t* = t(0.005, 4) ≈ 4.604
  
where t(0.005, 4) is the t-value with a cumulative probability of 0.005 and 4 degrees of freedom (found using a t-distribution table or calculator).
  
the margin of error for the confidence interval is then:
  
me = t* * (s / √n) = 4.604 * (3.46 / √5) ≈ 6.57
  
finally, the confidence interval for the difference in means is:
  
d̄ ± me = 4 ± 6.57
  
therefore, the 99% confidence interval for the difference in means μd = μ1 - μ2 using the paired data is approximately (-2.57, 10.57).

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Margin of Error + Confidence level

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