# Test of Hypothesis Confidence Interval for R-Coefficient

Test of Hypothesis Confidence Interval for R-Coefficient

Significance Estimations

1-    Ho: β                     =          8                      df=8

H1: β                     ≠          6

2-                                     =          0.05

3-    Tc                             =          β2– β2/SE(β2) ____ β2 (with cap) – β2 (simple) / SE(β2 (with cap)

4-    Critical Values          =          t 2

5-    Decision Rule

6-    Result

Interval Estimation

β2 – tα2 SE(β2)         ≤          β2        ≤         β2 + tα2 SE(β2)

X2 – X21α-2 SE(X2) ≤         X2         ≤         X22T

≤         t

Confidence Interval for R-coefficient

β1 and β2

CI for β2

Z          =          β2– β2/Se(β 2)

T          =          β2– β2/Se(β 2)

Pr (-tα2 ≤ t ≤ tα2)    = 1-α

Putting t value

Pr (-tα2 ≤ β22/Se(β2) ≤ tα2)    = 1-α

Rearranging

Pr [β2-tα2 Se(β2) ≤ β2 ≤ β2+ tα2 Se(β2)]           = 1-α

β2 ± tα2 Se(β2)

β1       =          Pr [β1-tα2 Se (β1) ≤ β1 ≤ β1+ tα2 Se(β1)] = 1-α

β1       ±           tα2 Se (β1)

CI for β1

9.6643 ≤ β1 ≤ 39.2448

24.4545 ± 2.306 (6.4138)

24.4545 ± 14.7902

CI for δ2

X2 = (n-2) δ2/ δ2

Pr (X21-α2 ≤ X2 ≤  X2 α2) = 1-α

Putting X2 Value

Pr [ (n-2) δ2/X2 ≤ δ2 ≤  (n-2) δ2/X2 1-α2] = 1-α

Interpretation

If we establish 95% confidence limits on δ2 and if we maintain a priori that these limits will include true δ2, we shall be right in the long run 95% of the time.

Hypothesis Testing

The confidence interval approach two-sided or two-tail test

Ho:           β2                                  =          0.3

H1:           β2                                  ≠          0.3

100 (1-α ) % CI for β2

β2 is 0.5091

Test of Hypothesis Confidence Interval for R-Coefficient

Decision Rule

Construct a 100 (1-α ) % CI for β2. If the β2 under Ho falls within this C.I do not reject Ho, but if it falls outside this interval region Ho.

One-Sided or One Tail Test

Ho:           β2                                  ≤          0.3

H1:           β2                                  >           0.3

Hypothesis Testing the Test of Significance approach

Testing the significance of regression coefficient the t-test

Broadly speaking, a test of significance is a procedure by which sample results are used to verify the truth or falsity of a null hypothesis.

T          =          β2– β 2/Se(β2)

Pr (-tα2 ≤ β2– β2/Se(β2) ≤ tα2)    = 1 -α

Pr [β2-tα2 Se (β2) ≤ β2 ≤ β2+ tα2 Se(β2)]           = 1 -α

Once again let us revert to our consumption income example we know that

β2        = 0.5091.

Se(β2)=0.0357,

df=8

if we assume  = 5% tα2 =2.306.

If we let

Ho: β2 = β2*=0.3 and

H1: β2≠0.3 becomes.

Pr(0.2177≤ β2 ≤ 0.3823)=0.95

The 95% C.I for  β2 under the hypothesis that β2 =0.3

T=0.5091-0.3/0.0357=5.857

Table t-test of significance decision rule

Type of Hypothesis  Ho: the null Hyp.  H1: the alternate Hyp.   Decision Rule reject Ho if

Two Tail                     β2 = β2*                        β2≠ β2*                                    | t | > tα2 df

Right Tail                   β2 ≤ β2*                       β2> β2*                                      t | > tα2 df

Left Tail                      β2 ≥ β2*                       β2< β2*                                      t | < tα2 df

Testing the significance of δ2: the X­2 test.

X2 = (n-2) δ2 / δ2

Table of A summery of X2 test

Ho: the null Hyp.                  H1: the alternate Hyp.         Critical region reject Ho if

δ2= δ20                                    δ2> δ20                                    df(δ2) / δ20 > X2  α df

δ2= δ20                                    δ2< δ20                                    df(δ2) / δ20 < X2 (1-α)df

δ2= δ20                                    δ2≠ δ20                                    df(δ2) / δ20 > X2 α/2 df

or df(δ2) / δ20 < X2(1-α )df

the “zero” Null hyp. “2-t”

Test of Hypothesis Confidence Interval for R-Coefficient

Rule of Thumb

“2-t” Rule of Thumb. If the number of df is 20 or more and if α , the level of significance is set at 0.05, then the null hyp. β2=0 can be rejected if the t value [=β2/se(β2)]

Ho: β2 = 0

T= β2/se(β2) >tα/2  when β2>0

T= β2/se(β2)<-tα/2  when β2<0

| t | = | β2/se(β2) < tα/2

| t | = | β2/se(β2) < tα 