X is a continuous random variable following uniform distribution between
3 and 10. We can write it as X~uniform (3,10). Please calculate the
probability Pr(X = 4) . (5%)
If we build a regression model to fit the weights of elementary school
students with age and height, and derive the regression line as follows:
weight = 10 + 0.2× age + 0.02× height . Which explanatory variable seems to
be more related to weights? (5%)
age
height
cannot tell.
We can approximate the distribution of a sample proportion with normal
distribution when the sample number is large. This is based on (5%)
Law of large number
Bayes’ Law
Central Limite Theorem
maximum likelihood
To know whether waitresses earn larger tips than waiters, we did a survey
with 50 randomly selected restaurants. We record the tips for one
randomly selected waiter and one randomly selected waitress in each of
the restaurant for a one-week period. The data can be summarized as
follows. If x represents data from 50 waiters and y represents data
from 50 waitresses, please answer our original question with significance
level 0.05. You need to state clearly your hypotheses and specify the
parameters of interests (15%)
There are three candidates, X,Y and Z for the coming election. During
a survey from mass media, 53%, 19.8%, 27.2% of the sampled voters will
vote for X , Y , and Z respectively. In general, 80%, 30%, 25% of
supporters for party A , B and C will vote candidate X , while 10%,
60% and 22% of supporters for party A , B and C will vote candidate
Y . If the sampled voters can all be grouped into the supporters for any one
of the three parties, what is the proportion of supporters for each of the
three parties in this survey? (10%)
(a) We want to design an opinion poll to estimate the public support for
party A. If we want the 95% confidence interval of the rate to be within
1% of the sample proportion (sample proportion ±1% ), what is the
minimum number of samples needed for this estimation? (10%)
(b) Suppose the public support for party A is 0.64 based on the record of
last year. If we conduct an opinion poll with 900 samples this year and
derive the supporting rate for party A as 0.6, please test if the public
support for party A has significantly decreased since last year. Use
significance level 0.05 for your test. (10%)
If the variances of two random variables X and Y are both σ 2 , what is
the maximum possible variance of their average? (7%)
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