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P1.T2.Ch.6 BT Notes (Hypothesis Testing)

dtammerz

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Hello

With regards to P1.T2 (QA) Ch.6 (Hypothesis Testing) Notes: in the "Identify the steps to test a hypothesis about the difference between two population means" (p.28-29)

where does the 0.78 and the 1.02 come from? I feel like i'm missing something.

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David Harper CFA FRM

David Harper CFA FRM
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#2
Hi @dtammerz Those are solutions to the maximum difference (hopefully the "⇒" is not throwing you off; "⇒" signifies "implies")

The denominator (i.e., the square root) is the standard deviation (aka, standard error) of the difference between correlated means and is equal to sqrt[to 4 + 1 - 2*0.3*sqrt(4)*sqrt(1)] = 0.398 (shown in exhibit).

So we really just have T = |D|/ σ(diff); i.e., the test-statistic T is the raw difference standardized by dividing by σ(diff). Just like we standardize the raw difference between the observed sample mean and the null hypothesized mean, X - μ, by dividing it by the SE, (X - μ)/SE, to retrieve the test statistic for a (univariate) sample mean.

Given T = |D|/ σ(diff), the max distance |D| = T*σ(diff); in this case, |D|= T*0.398. If we seek two-sided 95.0%, then |D| = 1.96*0.398 = 0.78, and if we seek two-sided 99.0% confidence, then |D| = 2.58*0.398 = 1.02. I hope that's helpful!
 

dtammerz

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Hi @dtammerz Those are solutions to the maximum difference (hopefully the "⇒" is not throwing you off; "⇒" signifies "implies")

The denominator (i.e., the square root) is the standard deviation (aka, standard error) of the difference between correlated means and is equal to sqrt[to 4 + 1 - 2*0.3*sqrt(4)*sqrt(1)] = 0.398 (shown in exhibit).

So we really just have T = |D|/ σ(diff); i.e., the test-statistic T is the raw difference standardized by dividing by σ(diff). Just like we standardize the raw difference between the observed sample mean and the null hypothesized mean, X - μ, by dividing it by the SE, (X - μ)/SE, to retrieve the test statistic for a (univariate) sample mean.

Given T = |D|/ σ(diff), the max distance |D| = T*σ(diff); in this case, |D|= T*0.398. If we seek two-sided 95.0%, then |D| = 1.96*0.398 = 0.78, and if we seek two-sided 99.0% confidence, then |D| = 2.58*0.398 = 1.02. I hope that's helpful!
Thank you David. that makes sense now. One more question; what exactly does "|D|" notation represent?
 

David Harper CFA FRM

David Harper CFA FRM
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#4
Sure @dtammerz The vertical bars represent absolute value (https://en.wikipedia.org/wiki/Absolute_value); as in, for example, the absolute value of |-2.33| = 2.33. As mentioned, the numerator given by "D" is the raw distance between the sample means; but it doesn't matter if the difference is positive or negative; the example shows a difference of µ(X) - µ(Y) = 0.75, but it wouldn't (and shouldn't) matter if we calculated a raw difference of µ(Y) - µ(X) = -0.75. Thanks,
 

dtammerz

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Sure @dtammerz The vertical bars represent absolute value (https://en.wikipedia.org/wiki/Absolute_value); as in, for example, the absolute value of |-2.33| = 2.33. As mentioned, the numerator given by "D" is the raw distance between the sample means; but it doesn't matter if the difference is positive or negative; the example shows a difference of µ(X) - µ(Y) = 0.75, but it wouldn't (and shouldn't) matter if we calculated a raw difference of µ(Y) - µ(X) = -0.75. Thanks,
Thank you!!
 
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