Question

Consider the following data drawn independently from normally distributed populations: Use Table 1.

  

x−1x−1 = 32.6	x−2x−2 = 27.8
σ12 = 89.5	σ22 = 93.4
n1 = 25	n2 = 22

   

a.	Construct the 90% 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.)

  

Confidence interval is  to .

    

b.	Specify the competing hypotheses in order to determine whether or not the population means differ.
	H0: μ1 − μ2 = 0; HA: μ1 − μ2 ≠ 0 H0: μ1 − μ2 ≥ 0; HA: μ1 − μ2 < 0 H0: μ1 − μ2 ≤ 0; HA: μ1 − μ2 > 0

  

c.	Using the confidence interval from part a, can you reject the null hypothesis?
	No, since the confidence interval includes the hypothesized value of 0. Yes, since the confidence interval does not include 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.

Answer 1

The variances do not differ a lot. Hence, we will calculate the confidence interval using the equal variances formula.The formula for the confidence interval is:

Sp or pooled variance can be calculated by:

Let's input all the values and calculate Sp.

x−1 = 32.6	x−2 = 27.8
σ12 = 89.5	σ22 = 93.4
n1 = 25	n2 = 22

Sp² = (24*89.5 + 21*93.5)/45 = 91.3667

t-critical value at 90% confidence interval and df = n1 + n2 - 2 = 45 is: 1.68

Using the formula of CI:

Lower bound: (32.6 - 27.8) - 1.6794*2.7942 = 4.8 - 4.6926 = 0.1074

Upper bound: (32.6 - 27.8) + 1.68*2.795 = 4.8 + 4.6926 = 9.4926

The confidence interval is: (0.11, 9.49)

b) The correct option is:

H₀: μ₁ − μ₂ = 0; H_A: μ₁ − μ₂ ≠ 0

c) The CI does not include 0, hence we can reject the null hypothesis. The correct options are:

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