# Problem Quants- Confidence Interval

Question from the curriculum:

Selected Regression Output - Dependent Variable: Amtex Share Return

Intercept 0.0095 ( coefficient );  0.0078 ( standard error)
Oil return 0.2354 ( coefficient ); 0.0760 ( standard error)
( Note: The critical t-value for a one-sided t-test at the 5% significance level
is 1.691)

28.  Vasileva should compute the:
A coefficient of determination to be 0.4689.
B 95% confidence interval for the intercept to be –0.0037 to 0.0227.
C 95% confidence interval for the slope coefficient to be 0.0810 to 0.3898.

The correct answer is C while I got B because I have used a df of 35 ( 36 total number of observations) and significance level of 5% ( t= 1.691 as shown above).

Thus, I did this for the confidence interval: 0.0095+ 1.691 ( 0.0078) = 0.22682 ( upper level)  and 0.0095-1.691 (0.0078)= -0.003682 (lower level)

C is correct. The confidence interval for the slope coefficient is calculated as:
Where b1 0.2354, sb1= 0.0760 and tc = 2.032

The lower limit for the confidence interval = 0.2354 – (2.032 × 0.0760) = 0.0810
The upper limit for the confidence interval = 0.2354 + (2.032 × 0.0760) = 0.3898

I don’t get it ; where is this 2.032 coming from and why are they not using 1.691 if it was given?

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How many observations?

What’s the confidence level or significance level for the confidence intervals?

Simplify the complicated side; don't complify the simplicated side.

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36 observations and 95% confidence level

How many degrees of freedom do you have?  (Hint: it isn’t 35.)

The note about the t-value is useless.  Why?

Simplify the complicated side; don't complify the simplicated side.

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thats what i am confused about. Why isnt it 35? and why did they give us the T value if its useless?

Onda wrote:
thats what i am confused about. Why isnt it 35?

How do you compute degrees of freedom?

Onda wrote:
and why did they give us the T value if its useless?

A better question is this: why didn’t you look up the correct t-value yourself?

Simplify the complicated side; don't complify the simplicated side.

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36-1=35

what i am also confused on , is that they gave us the information of the t- value being  1.691 for one side test. So again I was confused if I should look at alpha =0.05 or alpha = 0.025

Onda wrote:
36-1=35

Where did you get the “1”?

(Hint: it’s wrong.)

Simplify the complicated side; don't complify the simplicated side.

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n-1?

Onda wrote:
n-1?

No.

Simplify the complicated side; don't complify the simplicated side.

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can you tell me what exactly i should be doing? I get it that its degree of freedom, but why isnt it n-1 and second why were we given a t- value if its useless? am i looking at the one tail test

Onda wrote:
can you tell me what exactly i should be doing?

No.  I want you to discover that for yourself.  It’ll be more useful to you that way.

Onda wrote:
I get it that its degree of freedom, but why isnt it n-1 … ?

What does the “1” in the formula n − 1 represent?

Onda wrote:
… why were we given a t- value if its useless?

Got me.

Onda wrote:
am i looking at the one tail test

What do you mean by “test”?

(Hint: look at the title you chose for this thread.)

Simplify the complicated side; don't complify the simplicated side.

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I got it. The DF is n-k-1, so 34 is df  and the t value associated with it 2.032 (at either 0.05 alpha or 0.025 alpha)

Onda wrote:
I got it. The DF is n-k-1, so 34 is df …

Yes: you’ve estimated two parameters, so you’ve lost two degrees of freedom.

Onda wrote:
… and the t value associated with it 2.032 (at either 0.05 alpha or 0.025 alpha)

It’s not the case that α = 0.05 or α = 0.025.  It’s only one of those.

Which one?

How do you get t = 2.032?

Why?

Simplify the complicated side; don't complify the simplicated side.

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DF=34 and alpha =0.025 equal to t=2.032

Onda wrote:
DF=34 and alpha =0.025 equal to t=2.032

If C is correct, then α = 1 − confidence level = 1 − 95% = 5%, not 2.5%.

So, why is t = 2.032 correct?  (It is, by the way.)

Simplify the complicated side; don't complify the simplicated side.

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im terribly confused.

But isnt it because its a two tail test? So the whole alpha is 5%, but if we divide by two , each tail is 0.025

Onda wrote:
im terribly confused.

We’ll fix that.

Onda wrote:
But isnt it because its a two tail test?

You keep using the word “test”.  Why?

(As I mentioned before, look at the title you chose for this thread.)

Onda wrote:
So the whole alpha is 5% …

Yes.

Onda wrote:
… but if we divide by two, each tail is 0.025

But the area in each tail isn’t α, it’s … what?

Simplify the complicated side; don't complify the simplicated side.

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confidence interval is always a two tailed test

Onda wrote:
confidence interval is always a two tailed test

A confidence interval is two-tailed, but it’s not a test.

Don’t confuse confidence intervals with hypothesis testing.

Simplify the complicated side; don't complify the simplicated side.

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