Mathematical Statistics with Applications
Mathematical Statistics with Applications
7th Edition
ISBN: 9781111798789
Author: Dennis O. Wackerly
Publisher: Cengage Learning
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Chapter 11.14, Problem 88E

a.

To determine

State whether the given statement is true or false.

a.

Expert Solution
Check Mark

Answer to Problem 88E

The given statement is false.

Explanation of Solution

The statement is, if Model I is fit, the estimate for σ2 is based on 16 degrees of freedom.

Model I is as follows:

Y=β0+β1x1+β2x2+β3x3+β4x4+ε

An unbiased estimator of σ2 is as follows:

s2=SSEn(k+1)

According to this, the estimate for σ2 is based on n(k+1) degrees of freedom.

It is given that n=20

From Model I, the value of k=3.

The degrees of freedom is as follows:

n(k+1)=20(4+1)=15

This indicates that the estimate for σ2 is based on 15 degrees of freedom.

Since the estimate for σ2 is based on 15 degrees of freedom, the given statement is false.

b.

To determine

Explain whether the given statement is true or false.

b.

Expert Solution
Check Mark

Answer to Problem 88E

The given statement is true.

Explanation of Solution

The fitted Model II is as follows:

y^=β^0+β^1x1+β^2x2

To determine whether x2 contributes to a better fit of the model to the data, one needs to test the null hypothesis H0:β2=0 against Ha:β20.

The significance level is 0.01.

The test statistic is as follows:

t=β^3(sc33)

Where,

s=SSEn(k+1)=YYβ^XYn(k+1), β^=(XX)1XY

Here, k+1 is the number of unknown βi values and n is the number of data values.

Also, it is noticed that cii is the element in row (i+1) and column (i+1) of (XX)1.

Where, 0ik

The rejection region for this test is as follows:

RR:{|t|>tα/2}={t<tα/2 or t>tα/2}

Where, P(t<tα/2)=α/2

The value of tα/2 can be found using Student’s t-distribution with n2 degrees of freedom.

According to this, if |t|<t0.005, fail to reject the null hypothesis H0:β2=0. This indicates that x2 does not contribute to a better fit of the model to the data.

If |t|>t0.005, reject the null hypothesis H0:β2=0. It can be concluded that x2 contributes to a better fit of the model to the data.

Thus, the given statement is true.

c.

To determine

Check whether SSEISSEII when models I and II are both fit.

c.

Expert Solution
Check Mark

Answer to Problem 88E

The given statement is true.

Explanation of Solution

If there are k independent variables, then the fitted model is as follows:

y^=β^0+β^1x1+β^2x2+β^kxk

It is known that SSE=i=1n(yiy^i)2

Where y^i is the predicted value of yi and 1in

Here, Model I contains four independent variables x1, x2, x3 and x4.

Also Model II contains two independent variables x1 and x2.

This indicates that the predicted value for yi cannot be worse for Model I than Model II.

Where, 1i20

This means that (yiy^i)2 cannot be greater when Model I is used than Model II. Therefore, in this case, SSE cannot be greater for Model I than Model II. That is, SSEISSEII

According to this, the given statement is true.

d.

To determine

Check whether σ^2Iσ^2II when Models I and II are both fit.

d.

Expert Solution
Check Mark

Answer to Problem 88E

The given statement is false.

Explanation of Solution

It is known that an unbiased estimator for σ2 is as follows:

s2=SSEn(k+1) or σ^2=SSEn(k+1)

Where, SSE=i=1n(yiy^i)2, y^i is the predicted value of yi and 1in(=20).

Also k+1 is the number of unknown βi values and n is the number of data values.

For Model I, it is seen that k1=4 and for Model II, one can have k2=2.

Now, compute the values of σ^2I and σ^2II as follows:

σ^2I=SSEIn(k1+1)=SSEI15

σ^2II=SSEIIn(k2+1)=SSEII17

From Part (c), it is seen that SSEISSEII.  Hence it cannot be concluded that σ^2Iσ^2II.

Therefore, the given statement is false.

e.

To determine

Check whether Model II is a reduction of Model I.

e.

Expert Solution
Check Mark

Answer to Problem 88E

The statement istrue.

Explanation of Solution

Models I and II are as follows:

Y=β0+β1x1+β2x2+β3x3+β4x4+εY=β0+β1x1+β2x2+ε

It is observed that Model II is a subset of Model I.

That is, Model I=Model II+β3x3+β4x4+ε

Therefore, the given statement is true.

f.

To determine

Check whether Model III is a reduction of Model I.

f.

Expert Solution
Check Mark

Answer to Problem 88E

The given statement isfalse.

Explanation of Solution

Models I and III are as follows:

Y=β0+β1x1+β2x2+β3x3+β4x4+εY=β0+β1x1+β2x2+β3x1x2+ε

Here, Model III contains x1x2 and Model I do not contain x1x2. This indicates that Model III is not a reduction of Model I. Therefore, the given statement is false.

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Chapter 11 Solutions

Mathematical Statistics with Applications

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