Stat 407
Quiz 2
Name
1. (2pts) Which of the following samples is most likely to correspond to this correlation matrix
"
#
1 0.5
?
0.5 1
The middle plot because correlation between the two variables is 0.5.
2. (2pts) From this formula for a bivariate normal density tell me what the mean is and the variance of the
distribution is.
1
1 x1 − 10 2
x1 − 20 2
f (x1 , x2 ) = √
× exp −
( √
) +( √
)
2
2π 2 × 3
2
3
The mean is
"
10
10
#
and the variance is
"
2 0
0 3
#
3. (2pts) Based on the following SAS output, would you describe the variable LLENGTH as coming from a
normal distribution?
The UNIVARIATE Procedure
Variable: LLength
Moments
N
Mean
Std Deviation
Skewness
Uncorrected SS
Coeff Variation
Stem
51
50
49
48
47
46
45
24
4.90065934
0.16249809
-0.53298
577.002416
3.31584132
Sum Weights
Sum Observations
Variance
Kurtosis
Corrected SS
Std Error Mean
Leaf
8
0344679
013359
11999
#
1
7
6
5
3359
4
8
1
----+----+----+----+
Multiply Stem.Leaf by 10**-1
1
24
117.615824
0.02640563
-0.5642116
0.60732945
0.03316978
Boxplot
|
+-----+
*--+--*
+-----+
|
|
|
The distribution is slightly skewed, and bimodal, but it is hard to really tell whether it is a sample from a
normal because the sample is so small.
4. (3pts) Calculate the T 2 = "n(X̄#− µ0 )0 S−1
n−1 (X̄ − µ0 ) statistic from the following sample values given n = 3
9
and a hypothesized mean
5
X̄ =
"
8
6
#
−1
S
=
"
1/3 1/9
1/9 4/27
#
Would you reject the null hypothesis?
T 2 = 0.78(= 7/9). With a value so small we are unlikely to reject the null hypothesis.
5. (1pt) Which of the following is NOT an assumption underlying the contruction of the test statistic when
testing hypotheses about the multivariate mean (such as in the previous question)?
(a) Ho is true.
(b) The sample is taken by simple random sampling methods.
(c) The variance-covariance matrices are equal.
(d) The population takes a multivariate normal shape.
The answer is (c). There is only one variance-covariance matrix so there is not an assumption of equal
covariances. (a), (b) and (d) are all assumptions underlying the construciton of the test statistic.
2