A survey was made of 100 customers in a bookstore. Sixty of the 100
indicted they visited the store because of a newspaper advertisement. The
remainder had not seen the ad. A total of 40 customers made purchases; of
these customers, 30 had seen the ad. What is the probability that a person
who did not see the ad made a purchase? What is the probability that a
person who saw the ad made a purse? (10%)
In a certain factory, machines A, B,C, and D produce the same product.
Of the total production, machine A, B,C, and D produce 10%, 20%, 30%,
and 40%, respectively. The proportions of defective items produced by
machines A, B,C, and D are 0.1%, 0.05%, 0.5% and 0.2%, respectively.
An item selected at random is found to be defective. What is the item was
produced A ? by B ? by D ? (15%)
When choosing from a set of possible point estimators, a reasonable
approach is to choose the estimator with the smallest mean squared
error.
The standard error of a statistic is the standard deviation of the
statistic.
Problem (h.1) is a inferential statistics problem and Problem (h.2) is a
probability problem.
(h.1) A company can only tolerate 5% defective rate for its manufacturing
processes. Suppose that 200 items are inspected and 20 are
found to be defective, should we accept the process?
(h.2) If the defective rate is 5%, what is the probability that 200 items
are inspected and 20 or more are found to be defective?
Applying central limit theorem, the normal distribution can be used to
approximate the sampling distribution of the sample mean from a
random sample with a large sample size.
Applying central limit theorem, normality of the random samples
needs to be assumed.
Suppose that T follows a t distribution with degrees of freedom 10
and Z follows a standard normal distribution. P(T > 2) ≥ P(Z > 2) .
Suppose that we obtain a 95% confidence of the mean μ to be (65.5,
68.4). We know that the unknown mean satisfies 65.5 ≤ μ ≤ 68.4 .
Suppose that we obtain a 95% confidence of the mean μ to be (65.5,
68.4). We know that P(65.5 ≤ μ ≤ 68.4) = 0.95 .
Suppose that we obtain a 95% confidence of the proportion p to be
(0.5, 0.6). We know that if we repeat the experiment many times, in
the long run only 95% of the intervals would include the unknown
proportion p , but we don’t know whether (0.5, 0.6) includes p or not.
For any random variables X and Y , the following equation is true.
E(Y ) = E(E(Y |X))
For any random variables X and Y , the following equation is true.
V(Y ) = V(E(Y |X))
Suppose that voters, choosing between a Republican and a Democratic
candidate, give the Republican p× 100% of the votes. We take a
random sample of all voters. Consider the statistical hypotheses:
H0 : p ≤ 0.5 , H1 : p > 0.5 . Suppose that we take a sample of size n
and make a decision that we reject the null hypothesis. That means
that the testing-hypothesis approach has helped us to see the truth:
“Republican will win”.
Sum of two exponential random variable is also an exponential random
variable.
A gamma random variable is a special case of an exponential random
variable.
A Bernoulli random variable is a special case of a binomial random
variable.
Memoryless property P(X > m+ n|X > n) = P(X > m) holds for any
random variable X .
If X ~Normal distribution with mean μ and variance σ 2 , then μ
equals to the median of X .
Suppose X is a non-negative random variable. Then, Markov
Inequality holds: P(X ≥ a) ≤ E(X) a for all a > 0 .
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