Question

Consider the following data drawn independently from normally distributed populations: (You may find it useful to reference the appropriate table: z table or t table)

x−1x−1 = 27.7	x−2x−2 = 30.1
σ12 = 92.8	σ22 = 87.5
n1 = 24	n2 = 33

a. Construct the 99% confidence interval for the difference between the population means. (Negative values should be indicated by a minus sign. Round all intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)
  

b. Specify the competing hypotheses in order to determine whether or not the population means differ.
  

• H₀: μ₁ − μ₂ = 0; H_A: μ₁ − μ₂ ≠ 0

• H₀: μ₁ − μ₂ ≥ 0; H_A: μ₁ − μ₂ < 0

• H₀: μ₁ − μ₂ ≤ 0; H_A: μ₁ − μ₂ > 0

c. Using the confidence interval from part a, can you reject the null hypothesis?
  

• Yes, since the confidence interval does not include the hypothesized value of 0.

• No, since the confidence interval includes the hypothesized value of 0.

• Yes, since the confidence interval includes the hypothesized value of 0.

• No, since the confidence interval does not include the hypothesized value of 0.

d. Interpret the results at αα = 0.01.

• We conclude that the population means differ.

• We cannot conclude that the population means differ.

• We conclude that population mean 2 is greater than population mean 1.

• We cannot conclude that population mean 2 is greater than population mean 1.

Answer 1

A)

sample #1   ------->              
mean of sample 1,    x̅1=   27.7          
population std dev of sample 1,   σ1 =    9.633275663          
size of sample 1,    n1=   24          
                  
sample #2   --------->              
mean of sample 2,    x̅2=   30.1          
population std dev of sample 2,   σ2 =    9.354143467          
size of sample 2,    n2=   33          
                  
difference in sample means = x̅1 - x̅2 =    27.7   -   30.1   =   -2.4

Level of Significance ,    α =    0.01          
z-critical value =    Z α/2 =    2.5758   [excel function =normsinv(α/2) ]      
                  
std error , SE =    √(σ1²/n1+σ2²/n2) =    2.5531          
margin of error, E = Z*SE =    2.5758   *   2.553   =   6.5763
                  
difference of means = x̅1 - x̅2 =    27.7   -   30.1   =   -2.400
confidence interval is                   
Interval Lower Limit= (x̅1 - x̅2) - E =    -2.400   -   6.576   =   -8.9763
Interval Upper Limit= (x̅1 - x̅2) + E =    -2.400   +   6.576   =   4.1763
CI (-8.98 , 4.18)

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B)

• H₀: μ₁ − μ₂ = 0; H_A: μ₁ − μ₂ ≠ 0

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C)

• No, since the confidence interval includes the hypothesized value of 0.

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D)

• We cannot conclude that the population means differ.

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