Description
Problem 1. Given the dataset
ID RACE SBP DBP HR
001 W 130 80 60
002 B 140 90 70
003 W 120 70 64
004 W 150 90 76
005 B 124 86 72
(NOTE: SBP is systolic blood pressure, DBP is diastolic blood pressure, and HR is heart rate.)
Write an R code to compute the ”average” blood pressure. (ABP) defined as a weighted average of
the diastolic blood pressure and the systolic blood pressure. Since the heart spends more time in its
relaxed state (diastole), the diastolic pressure is weighted two-thirds, and the systolic blood pressure
is weighted one-third. Therefore, the average blood pressure could be computed by multiplying the
diastolic blood pressure by 2/3, and the systolic blood pressure by 1/3 and adding the two. An equivalent
expression would be the diastolic pressure plus one-third of the difference between the systolic and
diastolic pressures. Using either definition, add APB to the data set.
Problem 2. Rice (1995, p. 390) gives the following data (Natrella, 1963) on the latent heat of the
fusion of ice (cal/gm):
Method A: 79.98 80.04 80.02 80.04 80.03 80.03 80.04 79.97 80.05 80.03 80.02 80.00 80.02
Method B: 80.02 79.94 79.98 79.97 79.97 80.03 79.95 79.97
(a) Inspect the data graphically in various ways, for example, boxplots, Q-Q plots and histograms.
(b) Assuming normality, test the hypothesis of equal means, both with and without making the assumption of equal variances.
(c) Compare the result with a Wilcoxon/Mann-Whitney nonparametric two sample test.
[Remark: For (a), solve it both by paper and pencil and also using a computer. For (b) and (c), use
a computer for your calculation, but write out the corresponding formulas]
Problem 3. Given the following data points:
-1.43 -0.95 -0.19 0.02 0.14 0.83 1.35 1.46 2.62
(a) Calculate the sample median, IQR (interquartile range) and MAD (median absolute deviation);
What are the (robust) breakdown points of these statistics? (explain your answers)
(b) Draw a histogram, using 3 groups (with equal bandwidths).
[Remark: Solve this problem by paper and pencil.]
Problem 4. Consider the model yi = β0 + xiβi + i
, where i are iid N(0, σ2
) for i = 1, 2, . . . , 5. We
have the following data:
1
(a) Find the Least Squares (LS) and MLE estimates of β1, σ
2 and variance of βˆ
1.
(b) Find the 95% confidence interval for β1.
[Remark: Solve this problem by paper and pencil.]
Problem 5. Suppose X1, · · · , Xn
iid∼ Uniform[0, θ]. We would like to estimate and make inference for
θ.
(a) Find the MLE θbMLE for θ and compute the density of θbMLE.
(b) Let the underlying truth be θ = 1. Suppose we are only able to get n = 60 samples. Generate
your own observations x1, · · · , x60. Based on them we can do bootstrap and get the bootstrap MLE θb∗.
Bootstrap for N = 1000 times and plot the density of θb∗.
Compare the distribution of θbMLE and θb∗ according to the density plot you obtain. Are they symmetric
or asymmetric?
(c)Based on θbMLE and 1000 θb∗ you get in (b), find P(θbMLE = 1) and P(θb∗ = θbMLE).
(d) Based on the 1000 θb∗ obtained in (b), build the bootstrap 90% confidence interval for θ. Simulate
the data for N = 1000 times and get the 90% confidence interval for θ based on the θbMLE. Compare
these two intervals.
(e) This example shows that boostrap does poorly. In fact, try to prove P(θbMLE = 1) = 0 and
P(θb∗ = θbMLE) = 1 − (1 −
1
n
)
n
. You may use these two facts to check whether you have done correctly
in (c).
[Hint: In this example, the Bootstrap Central Limit Theorem does not hold. This is an example that
the bootstrap method fails to work.]
[Remark: For (a) and (e), solve it by paper and pencil. For (b),(c),(d), use a computer for your
calculation, and write down the formula and results.]
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