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Suppose you fit the model y =0+1x1+1x22+3x2+4x1x2+to n = 25 data points with the following results:

^0=1.26,^1= -2.43,^2=0.05,^3=0.62,^4=1.81s^1=1.21,s^2=0.16,s^3=0.26, s^4=1.49SSE=0.41 and R2=0.83

  1. Is there sufficient evidence to conclude that at least one of the parameters b1, b2, b3, or b4 is nonzero? Test using a = .05.

  2. Test H0: 尾1 = 0 against Ha: 尾1 < 0. Use 伪 = .05.

  3. Test H0: 尾2 = 0 against Ha: 尾2 > 0. Use 伪 = .05.

  4. Test H0: 尾3 = 0 against Ha: 尾3 鈮 0. Use 伪 = .05.

Short Answer

Expert verified
  1. At 95% confidence interval, it can be concluded that1=2=3=4=0

  2. At 95% confidence interval, it can be concluded that1=0 .

  3. At 95% confidence interval, it can be concluded that2=0 .

4. At 95% confidence interval, it can be concluded that 3鈮0 .

Step by step solution

01

Goodness of fit test

H0:1=2=3=4=0

Ha: At least one of the parameters1,2,3, and4is non zero

Here, F test statistic =SSEn-(k+1)=0.4125-5=0.0205

Value of F0.05,25,25 is 1.964

H0 is rejected if F statistic > F0.05,28,28. For 伪=0.05, since F < F0.05,28,28 Not sufficient evidence to reject Ho at 95% confidence interval.

Therefore, 1=2=3=4=0 .

02

Significance of β

H0: 1=0

Ha: 1< 0

Here, t-test statistic = ^1s^1=-2.431.21=-2.008

Value oft0.05,25is 1.708

H0 is rejected if t statistic > t0.05,25. For 伪=0.05, since t < t0.05,25 Not sufficient evidence to reject Ho at 95% confidence interval.

Therefore, 1=0 .

03

Significance of β3

H0:2= 0

Ha:2> 0

Here, t-test statistic = ^2s^2=0.050.16=0.3125

Value oft0.05,25is 1.708

H0is rejected if t statistic > t0.05,25. For 伪 = 0.05, since t < t0.05,31Not sufficient evidence to rejectHoat 95% confidence interval.

Therefore, 2=0 .

04

Significance of β3

H0:3=0

Ha: 3鈮0

Here, t-test statistic =^3s^3=0.620.26=2.38461

Value oft0.025,25is 2.060

H0is rejected if t statistic > t0.05,24,24. For 伪 = 0.05, since t>t0.05,31 Sufficient evidence to rejectHoat 95% confidence interval.

Therefore, 3鈮0 .

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