Midterm Exam with Practice Problems - Applied Multivariate Analysis | ISQS 6348, Exams of Descriptive statistics

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ISQS 6348 Midterm | Spring 2000

Instructions: Open book and notes. Points (out of 100) in parentheses.

1. Data vectors X 1 ; X 2 ; : : : ; X 25 are to be sampled from a population whose mean vector and covariance matrix are

¹ =

μ (^) ¹ 1 ¹ 2

¶

and § =

μ (^) ¾ 11 ¾ 12 ¾ 12 ¾ 22

¶

:

1.a.(2) Find E(X 2 ). (hint: it is identical to E(X 1 )). 1.b.(3) Find Cov(X 25 ). (hint: it is identical to Cov(X 1 )). 1.c.(2) Let a^0 = ( : 5 :5 ) for the remainder. Find E(a^0 X 1 ). 1.d.(3) Find Var(a^0 X 1 ). 1.e.(3) Let X¹ = 251 §^25 i=1Xi. Find E(a^0 X¹). 1.f.(5) Find Var(a^0 X¹). 1.g.(2) Find the standard deviation of a^0 X¹.

2.(20) Suppose you now have carried out the study in question 1, and you have collected the data. Suppose the sample mean vector and sample covari- ance matrix are

X¹ =

μ (^30) : 0 10 : 0

¶

and S =

μ (^16) : 0 4 : 0 4 : 0 4 : 0

¶

:

Find the 95% con¯dence interval for (¹ 1 + ¹ 2 )=2.

3. An experiment yields the following (unadjusted, or ordinary) p-values: For hypothesis H 1 , p 1 = :013, for hypothesis H 2 , p 2 = :49, for hypothesis H 3 , p 3 = :011, for hypothesis H 4 , p 4 = :50. 3.a.(10) Calculate the Bonferroni adjusted p-values and state which hypothe- ses are rejected. State also what signi¯cance level you are applying, and state whether it refers to EER or CER.

3.b.(10) Calculate the step-down Bonferroni adjusted p-values and state which hypotheses are rejected. State what signi¯cance level you are ap- plying, and state whether it refers to EER or CER.

4. Assume the distribution of X is multivariate normal with mean vector and covariance matrix

¹ =

0

@

0

0

0

A ; § =

0

@

A :

The eigenvalue, eigenvector pairs of § are

¸ 1 = 4; e 1 =

0

@

0

0

A (^) ; ¸ 2 = 16; e 2 =

0

@

0

0

A (^) ; ¸ 3 = 1; e 3 =

0

@

0

0

A :

4.a.(10) Find the Euclidean length of the major axis of the 95% probability ellipsoid. 4.b.(10) Describe the appearance of the ellipsoid: How does it look?

5. A sample of 100 ¯nancially viable ¯rms and a sample of 50 ¯rms with ¯nancial problems are compared. Variables include X 1 = (Debt ratio), X 2 = (Assets to liabilities ratio). Both measures are ratios with reason- ably common scales (or variances). The most signi¯cant linear combination is MSLC= ¡: 04 X 1 + : 28 X 2. 5.a.(5) Interpret the linear combination in terms of what it DOES. 5.b.(5) Interpret the linear combination in terms of what it MEASURES. 5.c.(10) Find the appropriate critical value for the two-sample t-statistic based on the MSLC in this example. (Emphasis on the word appropriate.)