The following are the final exam scores of 13 students in a Statistics course 142,
132, 116, 163, 118, 123, 124, 156, 124, 121, 179, 124, 133
(1) Present the distribution of the scores using a stem-and-leaf plot. (5%)
(2) Are there any suspected outliers according to the 1.5 IQR criteria? (10%)
(3) Base on the shape of the distribution, would you report the mean or the
median as a measure of the center? Explain your choice. (5%)
A friend tells you he has a coin that lands heads 30%, 50%, or 70% of the time,
but he does not tell you which. Before seeing any data you assume that each
possibility is equally likely. He then lets you toss the coin 8 times, and it lands
heads 7 times.
(1) What is your posterior probability that p = 0.5 ? (10%)
(2) What is your prediction for the probability that the coin will land heads the
nest time it is tossed? (5%)
(3) Suppose you wanted to test 0 : 0.5 H p = against 1 : 0.5 H p > . What is the
observed p-value? (5%)
A binomial experiment is based on 100 trials and an unknown success
probability p. Let X is the number of successes. The null hypothesis is
H0 : p = 0.5 and the alternative hypothesis is Hi : p = 0.6 . Hi is accepted if
X > 58. Find the probabilities of type 1 and type 2 errors. Use the normal
approximation and the continuity correction. [Note:標準差取到小數點一位即
可]
Mr. Bob concludes that according to the coefficient estimates, X2t is the
most important explanatory variables among others. Is his conclusion
correct? Why or why not? (10%)
Suppose all the assumptions for obtaining the best linear unbiased
estimators are satisfied in this case. Moreover, suppose that the true
coefficient of X1t is 0.20, which is, of course, unknown to Mr. Bob.
Based on the above estimation results, what is the power of hypothesis test
that there is no effect of X1t on Y ? (Approximate the probability by the
normal distribution). (10%)
State all the assumptions needed for Mr. Bob in question (1) to get the best
linear unbiased estimators by the ordinary least squares method. (10%)
Indicate the likely violation of the assumptions in question (1).
How would you evaluate if a given data series follows the normal distribution?
(10%)
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