Consider the following data: Use Table 1. x_1 = 30.8 x_2 = 25.3 sigma_1^2 = 95.4

Question

Consider the following data: Use Table 1. x_1 = 30.8 x_2 = 25.3 sigma_1^2 = 95.4 sigma_2^2 = 96.6 n_1 =25 n_2 = 28 Construct a 95% confidence interval for the difference between the population means. (Negative values should be indicated by a minus sign. Round intermediate calculations to 4 decimal places and final answers to 2 decimal places.) Specify the competing hypotheses in order to determine whether or not the population means differ. H_0: mu_1 - mu_2 lessthanorequalto 0; H_A: mu_1 - mu_2 > 0 H_0: mu_1 - mu_2 = 0; H_A: mu_1 - mu_2 notequalto 0 H_0: mu_1 - mu_2 Greaterthanorequalto 0; H_A: mu_1 - mu_2

Explanation / Answer

a)

I will be using z distribution as sigma values are known, and n1 + n2 > 30.

Calculating the means of each group,              
              
X1 =    30.8          
X2 =    25.3          
              
Calculating the standard deviations of each group,              
              
s1 =    9.767292358          
s2 =    9.8285299          
              
Thus, the standard error of their difference is, by using sD = sqrt(s1^2/n1 + s2^2/n2):              
              
n1 = sample size of group 1 =    25          
n2 = sample size of group 2 =    28          

Also, sD =    2.695551892          
              
For the   0.95   confidence level, then      
              
alpha/2 = (1 - confidence level)/2 =    0.025      
Hence,
  
z(alpha/2) =    1.959963985          

Thus,
              
lower bound = [X1 - X2] - z(alpha/2) * sD =    0.216815374          
upper bound = [X1 - X2] + z(alpha/2) * sD =    10.78318463          
              
Thus, the confidence interval is              
              
(   0.216815374   ,   10.78318463   ) [ANSWER]

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b)

Formulating the null and alternative hypotheses,              
              
Ho:   u1 - u2   =   0  
Ha:   u1 - u2   =/   0   [ANSWER, B]

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c)

As the interval does not include 0, we reject Ho.

Hence, OPTION B. [ANSWER]

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Hi! If you use another convention is choosing between z and t distribution in your class, please resubmit this question together with your preferred method/distribution for this problem. That way we can continue helping you! Thanks!

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