Problem 1 At a Cola factory, the machine that is used to fill the bottles has been observed tohave a normal distribution. The true standard deviation in the amounts of fill is approximately
σ = 1.0 ounces. However, the mean ounces of fill μ may change from day to day, because of
change of operator or adjustments in the machine. If we measure the ounces of fill of n = 9
randomly selected bottles, then the probability that the sample mean will be within 0.35 ounce
of the true mean is 0.7063. Suppose that Y is to be computed using a sample of size n.
(a) If n = 16, what is P(|Y – μ| ≤ 0.35)?
(b) Find n for which P(|Ÿ – µ| ≤ 0.35) = 0.9643?
(c) If n = 49, what is P(|Ỹ – µ| ≤ 0.35)?
(d) Find n for which P(|Ÿ – µ| ≤ 0.35) = 0.9949.
(e) What pattern do you observe among the value of P(|Ÿ – µ| ≤ 0.35) that you observed for
the various value of n?
–
(f) Suppose that we take n = 11 samples (of the bottles with mean μ and sigma = 1.0). Use
the Chi-Square Probabilities and Quantiles to calculate the probability that the sample
variance S² of the fill of these bottles is at least 2.7 ounces. (Recall the result on the
distribution of the sample variance). You will need to use the applet we saw in class
applet.
(g) Assume the amount of fill is no longer normally distributed, but is some random variable
with mean μ and standard deviation σ = 1.25. What is the probability that the average
fill of 25 bottles is within 0.5 ounce of the true mean?Problem 2 Let Y₁, Y2,…, Yn denote a random sample of size n from a population whose density
is given by
B≤y,
f(y)
=
0, otherwise,
where > 0 is unknown. We want to compare two estimators for ẞ. Let ẞ₁ = min(Y₁, Y2,…, Yn).
(a) Derive the bias of the estimator B₁. Is ẞ₁ an unbiased estimator for ẞ? If not, derive an
unbiased estimator from ẞ₁, denote it by B’₁.
1
(b) Derive MSE(B’₁) (ps1: it can simply be that B’₁ = B₁ if the answer to question (a) is the
estimator is unbiased; ps2: if you failed to answer the question, calculate MSE(B1).
(c) Now let 32 = another estimator for ẞ. Is ẞ2 an unbiased estimator?
(d) Which of B1 and B2 is a better estimator? why? (the answer requires proper justification
from week 4 material)
Nb from 120A: Having an infinite variance is not chocking.
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