Page 277 - Applied statistics and probability for engineers

Section 7-3/General Concepts of Point Estimation 255

Exercises FOR SECTION 7-3

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.

7-22. A computer software package calculated some numer- Which estimator is better and in what sense is it better? Calcu-
ical summaries of a sample of data. The results are displayed here: late the relative eficiency of the two estimators.
∧ ∧
7-29. Suppose that Q1 and Q2 are estimators of the parameter θ.
∧
∧
∧
=
Variable N Mean SE Mean StDev Variance We know that E(Q1 = θ ,E(Q2 ) = θ /2 ,V(Q1 ) 10 , V(Q2 = ∧ ) 4.
)
x 20 50.184 ? 1.816 ? Which estimator is better? In what sense is it better?
∧ ∧ ∧
7-30. Suppose that Q1, Q2, and Q3 are estimators of θ. We know
∧
∧
∧
(a) Fill in the missing quantities. that E(Q1 = E(Q2 = θ , E(Q3 ≠ θ ∧ ) 12= ∧ ) 10= and
)
)
)
(b) What is the estimate of the mean of the population from E(Q3− θ = ,V(Q1 V , (Q2
∧
2
which this sample was drawn? ) 6. Compare these three estimators. Which do you
prefer? Why?
7-23. A computer software package calculated some
7-31. Let three random samples of sizes n 1 = 20, n 2 = 10,
numerical summaries of a sample of data. The results are and n 3 =
displayed here: 2 8 be taken from a population with mean μ and vari-
2
2
2
ance σ . Let S 1 , S 2 , and S 3 be the sample variances. Show that
2
2
2
2
2
SE Sum of S = ( 20 S 1 + 10 S 2 + 8 S 3 ) / 38 is an unbiased estimator of σ .
Variable N Mean Mean StDev Variance Sum Squares 7-32. (a) Show that ∑ n (X i − X ) / n is a biased estima-
2
2 i =1
x ? ? 2.05 10.25 ? 3761.70 ? tor of σ .
(b) Find the amount of bias in the estimator.
(a) Fill in the missing quantities. (c) What happens to the bias as the sample size n increases?
(b) What is the estimate of the mean of the population from 7-33. Let X , X , … be a random sample of size n
, X n
1
2
which this sample was drawn? from a population with mean μ and variance σ .
2
7-24. Let X 1 and X 2 be independent random variables with (a) Show that X is a biased estimator for μ .
2
2
2
mean μ and variance σ . Suppose that we have two estimators (b) Find the amount of bias in this estimator.
of μ:
(c) What happens to the bias as the sample size n increases?
∧ X 1 + X ∧ X 1 3+ X
Q1 = 2 and Q2 = 2 7-34. Data on pull-off force (pounds) for connectors used
2 4 in an automobile engine application are as follows: 79.3, 75.1,
(a) Are both estimators unbiased estimators of μ? 78.2, 74.1, 73.9, 75.0, 77.6, 77.3, 73.8, 74.6, 75.5, 74.0, 74.7,
(b) What is the variance of each estimator? 75.9, 72.9, 73.8, 74.2, 78.1, 75.4, 76.3, 75.3, 76.2, 74.9, 78.0,
1 ,
7-25. Suppose that we have a random sample X X 2 ,… , ∧ X n 75.1, 76.8.
from a population that is N( ,μ È 2 ). We plan to use Q = (a) Calculate a point estimate of the mean pull-off force of all
∑ i ( X i − ) 2 2 ∧ connectors in the population. State which estimator you
n
X / c to estimate σ . Compute the bias in Q as an
=1
2
estimator of σ as a function of the constant c. used and why.
7-26. Suppose we have a random sample of size 2n from a pop- (b) Calculate a point estimate of the pull-off force value that
2
ulation denoted by X, and E X ( ) = μ and V X ( ) = σ . Let separates the weakest 50% of the connectors in the popula-
1 2 n 1 n tion from the strongest 50%.
X 1 = ∑ X i and X 2 = ∑ X i (c) Calculate point estimates of the population variance and
2 n i 1 n i 1
=
=
the population standard deviation.
be two estimators of μ. Which is the better estimator of μ? (d) Calculate the standard error of the point estimate found in
Explain your choice. part (a). Interpret the standard error.
…
7-27. Let X , X , , X 7 denote a random sample from a (e) Calculate a point estimate of the proportion of all connectors
2
1
2
population having mean μ and variance σ . Consider the fol- in the population whose pull-off force is less than 73 pounds.
lowing estimators of μ:
∧ X 1 + X 2 + … + X 7-35. Data on the oxide thickness of semiconductor
Q1 = 7 wafers are as follows: 425, 431, 416, 419, 421, 436, 418, 410,
7 431, 433, 423, 426, 410, 435, 436, 428, 411, 426, 409, 437,
∧
Q2 = 2X 1 − X 6 + X 4 422, 428, 413, 416.
2
(a) Calculate a point estimate of the mean oxide thickness for
(a) Is either estimator unbiased? all wafers in the population.
(b) Which estimator is better? In what sense is it better? (b) Calculate a point estimate of the standard deviation of
Calculate the relative eficiency of the two estimators. oxide thickness for all wafers in the population.
∧ ∧
7-28. Suppose that Q1 and Q2 are unbiased estimators (c) Calculate the standard error of the point estimate from
∧ ∧
=
)
of the parameter θ. We know that V(Q1 ) 10 and V(Q2 = 4. part (a).