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statistics for engineers and scientists

Questions and Answers of Statistics For Engineers And Scientists

10. (a) Describe the variance and standard deviation. (b) Explain why the standard deviation is more often used as a descriptive statistic than the variance.
9. (a) Describe and explain the difference between the mean, median, and mode.(b) Make up an example (not in your lectures) in which the median would be the preferred measure of central tendency.
4. Here are the noon temperatures (in degrees Celsius) in a particular Canadian city on Thanksgiving Day for the 10 years from 2002 through 2011:0, 3, 6, 8, 2, 9, 7, 6, 4, 5. Describe the typical
3. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:2.13, 6.01, 3.33, 5.78
2. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:6, 1, 4, 2, 3, 4, 6, 6
1. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:32, 28, 24, 28, 28, 31, 35, 29, 26
22. Mouradian (2001) surveyed college students selected from a screening session to include two groups: (a) “Perpetrators”—students who reported at least one violent act (hitting, shoving,
21. Raskauskas and Stoltz (2007) asked a group of 84 adolescents about their involvement in traditional and electronic bullying. The researchers defined electronic bullying as “... a means of
20. Find an example in a newspaper, magazine, or news website of a graph that misleads by failing to use equal interval sizes or by exaggerating proportions.
19. Make up and draw an example of each of the following distributions: (a) bimodal,(b) approximately rectangular, and (c) skewed to the right. Do not use an example given in class.
18. Explain to a person who has never taken a course in statistics the meaning of a grouped frequency table.
17. Pick a book and a page number of your choice. (Select a page with at least 30 lines; do not pick a textbook or any book with tables or illustrations.) Make a list of the number of words on each
16. Here are the number of holiday gifts purchased by 25 families randomly interviewed at a local mall at the end of the holiday season:22, 18, 22, 26, 19, 14, 23, 27, 2, 18, 28, 28, 11, 16, 34, 28,
15. Following are the speeds of 40 cars clocked by radar on a particular road in a 35-mph zone on a particular afternoon:30, 36, 42, 36, 30, 52, 36, 34, 36, 33, 30, 32, 35, 32, 37, 34, 36, 31, 35,
14. A social psychologist asked 15 college students how many times they “fell in love” before they were 11 years old. The numbers of times were as follows:2, 0, 6, 0, 3, 1, 0, 4, 9, 0, 5, 6, 1,
13. An organizational psychologist asks 20 employees in a company to rate their job satisfaction on a 5-point scale from 1 = very unsatisfied to 5 = very satisfied.The ratings are as follows:3, 2, 3,
12. Explain and give an example for each of the following types of variables:(a) equal-interval, (b) rank-order, (c) nominal, (d) ratio scale, (e) continuous;(f) discrete.
11. A participant in a cognitive psychology study is given 50 words to remember and later asked to recall as many of the words as she can. She recalls 17 words.What is the (a) variable, (b) possible
8. Make up and draw an example of each of the following distributions: (a) symmetrical,(b) rectangular, and (c) skewed to the right.
6. The following data are the number of minutes it took each of a group of 34 10-year-olds to do a series of abstract puzzles:24, 83, 36, 22, 81, 39, 60, 62, 38, 66, 38, 36, 45, 20, 20, 67, 41, 87,
5. These are the scores on a test of sensitivity to smell taken by 25 chefs attending a national conference:96, 83, 59, 64, 73, 74, 80, 68, 87, 67, 64, 92, 76, 71, 68, 50, 85, 75, 81, 70, 76, 91, 69,
4. Fifty students were asked how many hours they studied this weekend. Here are their answers:11, 2, 0, 13, 5, 7, 1, 8, 12, 11, 7, 8, 9, 10, 7, 4, 6, 10, 4, 7, 8, 6, 7, 10, 7, 3, 11, 18, 2, 9, 7, 3,
2, 4, 2, 1, 0, 3, 6, 0, 1, 1, 2, 3, 2, 0, 1, 2, 1, 0, 2, 2 Make (a) a frequency table and (b) a histogram. Then (c) describe the general shape of the distribution.
3. A particular block in a suburban neighborhood has 20 households. The number of children in these households is as follows:
2. Give the level of measurement for each of the following variables: (a) ethnic group to which a person belongs, (b) number of times a mouse makes a wrong turn in a laboratory maze, and (c) position
1. A client rates her satisfaction with her vocational counselor as a 3 on a 4-point scale from 1 = not at all satisfied to 4 = very satisfied. What is the (a) variable,(b) possible values, and (c)
In the 4 x 4 Latin square of the examples, show that the nine degrees of freedom for (af3) interaction are being divided into three degrees of freedom for "Y and six degrees of freedom for error.A
Derive the analysis for the Graeco-Latin square given below. Use the model Yhijk = J.L + ah + f3i + "Yj + 17k + ehijk·CI C2 C3 C4 C5 RI TITI T2T3 T3T5 T4T2 T5T4 R2 T2T2 T3T4 T4TI T5T3 TIT5 R3 T3T3
For model (1), C(Mr) is given by Proposition 4.2.3.What is C(Mr) in the notation of model (2)?
Show that the contrasts in the TijS corresponding to the contrasts E Aiai and E E Ai''lj (a{3)ij are E E AiTij and E E Ai''ljTij, respectively.Hint: Any contrast in the TijS corresponds to a vector
Show that the set of indices i = 1, ... ,a, j = 1, ... , a, and k = (i + j + a - 1) mod(a) determines a Latin square design.Hint: Recall that tmod(a) is t modulo a and is defined as the remainder
Consider a one-way ANOVA with one covariate. The model is Yij = J-L + ai + t;Xij + eij, i = 1, ... , t, j = 1, ... , Ni . Find the BLUE of the contrast L~=l Aiai. Find the variance of the contrast.
Suppose >"i(3 and >"~'Y are estimable in model (9.0.1).Use the normal equations to find find least squares estimates of >..i(3 and>"~'Y.Hint: Reparameterize the model as X(3 + Z'Y = X8 + (/ - M)Z'Y
Derive the test for model (9.0.1) versus the reduced model Y = X(3 + Zo'Yo +e, where C(Zo) c C(Z). Describe how the procedure would work for testing Ho : 'Y2 = 0 in the model Yij = Jl. + Q!i + 1}j
An experiment was conducted with two treatments.There were four levels of the first treatment and five levels of the second treatment. Besides the data y, two covariates were measured, Xl and X2'The
Show that ~' T is estimable if and only if ~' T is a contrast.Hint: One direction is easy. For the other direction, show that for ~' =(6, ... ,~t),( = (k/)..t)(Z'(I - M)Z.
Show that if ~'T and 11'T are contrasts and that if~'11 = 0, then ~'T = 0 and r/T = 0 put orthogonal constraints on C(X, Z), i.e., the treatment sum of squares can be broken down with orthogonal
Derive the analysis for a Latin square design with one row missing.Hint: This problem is at the end of Section 9.4, not Section 9.3.
Eighty wheat plants were grown in each of five different fields. Each of six individuals (A, B, C, D, E, and F) were asked to pick eight "representative" plants in each field and measure the plants's
Sulzberger (1953) and Williams (1959) have examined the maximum compressive strength parallel to the grain (y) of 10 hoop trees and how it was affected by temperature. A covariate, the moisture
Suppose that in Exercise 7.7.1 on motor oil pricing, the observation on store 7, brand H was lost. Treat the stores as blocks in a randomized complete block design. Plug in an estimate of the missing
The missing value procedure that consists of analyzing the model (Y - Zi') = X{3 + e has been shown to give the correct SSE and BLUEs; however, sums of squares explained by the model are biased
State whether each design given below is a balanced incomplete block design, and if so, give the values ofb, t, k, r, and A.(a) The experiment involves five treatments: A, B, C, D, and E. The
Show that AoY is a BLUE of X,8 if and only if, for every estimable function)..',8 such that pi X = )..', pi Ao Y is a BLUE of )..',8.
Show that if A' = pi X and if AoY is a BLUE of X,8, then pi AoY is a BLUE of A',8.
The BLUE of X{J can be obtained by taking T =V + XX'. Prove this by showing that(a) C(X) C C(T) if and only if TT-X = X and(b) if T = V + XX', then TT- X = X.Hint: Searle and Pukelsheim (1987) base a
Show that V{(J - M)VI is invertible.Answer: Since the columns of VI form a basis, then 0 = VI b iff b = O.Also (I - M)Vlb = 0 iff Vlb E C(X), but Vib E C(X) iff b = 0 by choice of VI' Thus, (1 -
C(Vl'o) c C(l'o).Hint: Show that Show that if MY - MVli is a BLUE of Xj3, then[M - MVI [V{(I - M)Virl V{(J - M)] V= V [M - (J - M)VI [V{(J - M)VI]-1 V{M] .Then multiply on the right by Yo.We include
Give the general form for a BLUE of X{3 in model(10.1.1).Hint: Add something to a particular BLUE.
From inspecting Figure 10.2, give the least squares CLUE for Example 10.2.2. Do not do any matrix manipulations.
Show that ordinary least squares estimates are best linear unbiased estimates in the model Y = X{3 +e, E(e) = 0, Cov(e) = V if the columns of X are eigenvectors of V.
Use Definition B.31 and Proposition 10.4.6 to show that Mc(x)nc(v) = M - Mw where C(W) = C[M(J - Mv)].
The usual model for a randomized complete block design was given in Section 8.2 as Yij = /-L + Gi + (Jj + eij, i = 1, ... ,a, j = 1, ... ,b, Var(eij) = a 2, and Cov(eij, eiljl) = ° for (i,j)
An alternative model for a block design is Yij = /-L + Gi + (Jj + eij, (5)where the (JjS are independent N(O, a~) and the (JjS and eijS are independent. If this model is used for a balanced
Show that when using model (5) for a randomized complete block design, the BLUE of a contrast in the (tiS is the same regardless of whether the {3jS are assumed random or fixed. Show that the
(a) Give a detailed proof for the distribution of the statistic for testing model (8) against model (2).(b) Give a detailed proof for the distribution of the statistic for testing model (9) against
Consider the table of means T 1 2 m 1 Y·ll Y·12 Y·lm W 2 jj.21 jj.22 jj.2m t jj.tl jj.t2 jj.tm Let E;:=l dk = O. For any fixed j, find a confidence interval for where J.Ljk = J.L + (. + Wj + rk +
Prediction in Standard Linear Models.Our usual linear model situation is that the YiS have zero covariance and identical variances. Thus, model (2) is satisfied with V = (]'2 I. A new observation yo
Spatial Data and Kriging.If Yi is an observation taken at some point in space, then x~ consists of the coordinates of the point, or, more generally, some function of the coordinates. In dealing with
Prove three facts about Kriging.(a) Show that if b'Y is the BLUP of yo and if Xi! = 1, i = 0,1, ... , n, then b'J = 1.(b) Show that if (yO, x~) = (Yi, x~) for some i ~ 1, then the BLUP of yo is just
If >'1(3 is estimable, find the best linear unbiased predictor of >'1(3+>'~'Y. For this problem, bo+b'Y is unbiased ifE(bo+b'Y) =E(>'l(3 + >'~'Y). The best predictor minimizes E[bo + b'Y - >'1(3 -
Assuming the results of Exercise 6.3, show that the BLUP of the random vector A''Y is Q'(Y - X,6), where VQ = ZDA.
Consider the model Y = Xj3 +e, e ,...., N(O, a2 J).Show that the M S E is the REML estimate of a2 .
Consider the Method 3 estimates of 0'5 and O'~ in Example 12.9.1.(a) Show that these are also the REML estimates.(b) Show that the vector (Y'Y, Y'Mz Y, J'Y)' is a complete sufficient statistic for
1. Prove that r(qZt} = reX, Zt} - reX) when C(Zt} c C(X, Z2).2. Prove that qZ2Z~C2 is positive definite.3. Prove (6).4. Prove (7).
Use the data and model of Exercise 12.7 to test Ho : O'I = 0 and Ho : O'~ = O.
Show that for a regression model that does not contain an intercept, the diagonal elements of the perpendicular projection operator are equivalent to the estimated squared Mahalanobis distances
Show that the sample correlation between two vectors in the same direction is 1.
Using the model of Example 13.2.2, estimate the power of detecting a t(3) with a = .05.Hint: Do a simulation.
Show that d is approximately equal to 2(1 - r a ), where ra is the sample (auto)correlation between the pairs (ei+l' ei) i =1, ... ,n-1.
Show that the standardized predicted residuals with(j2 estimated by M SE are the same as the standardized residuals.
The data given below were first presented by Brownlee (1965) and have subsequently appeared in Daniel and Wood (1980), Draper and Smith (1981), and Andrews (1974), among other places. The data
For testing whether one observation Yi is an outlier, show that the F statistic is equal to the squared standardized predicted residual.
Consider testing the regression model (1) against(2). Show that F > 1 if and only if the Adj R2 for model (1) is less than the Adj R2 for model (2).
Give an informal argument to show that if Y =Xo'Y + e is a correct model, then the value of Cp should be around p.Provide a formal argument for this fact. Show that if (n - s) > 2, then E(Cp ) = p +
Consider the F statistic for testing model (1) against model (14.0.1): (a) show that Cp = (s - p)(F - 2) + Sj (b) show that, for a given value of p, the R2, Adj R2, and Cp criteria all induce the
Show that any linear combination of ill-defined orthonormal eigenvectors is ill-defined. In particular, if w = X(avi + bVj), then
Prove Theorem 14.4.3.Hint: For a design matrix with columns of length one, tr(X' X) = p. It follows that 1 ~ m~ 8i ~ p.
Which of the following equalities are valid: C(A) =C(A, D), C(D) = C(A, B), C(A, N) = C(A), C(N) = C(A), C(A)C(F), C(A) = C(G), C(A) = C(H), C(A) = C(D)?
Which of the following matrices have linearly independent columns: A, B, D, N, F, H, G?
Give a basis for the space spanned by the columns of each of the following matrices: A, B, D, N, F, H, G.
Give the ranks of A, B, D, E, F, G, H, K, L, N.
Which of the following matrices have columns that are mutually orthogonal: B, A, D?Exercise A.I0 Give an orthogonal basis for the space spanned by the columns of each of the following matrices: A, D,
Find two linearly independent vectors in the orthogonal complement of C(D) (with respect to R 4 ).
Find a vector in the orthogonal complement of C(D)with respect to C(A).
For X as above, find two linearly independent vectors in the orthogonal complement of C(X) (with respect to R 6 ).
Let X be an n x p matrix. Prove or disprove the following statement: every vector in Rn is in either C(X) or C(X)J.. or both.
For any matrix A, prove that C(A) and the null space of A' are orthogonal complements. Note: The null space is defined in Definition B.ll.
Let Ml and M2 be perpendicular projection matrices, and let Mo be a perpendicular projection operator onto C(M1 ) n C(M2). Show that the following are equivalent:(a) MIM2 = M2M1·(b) MIM2 = Mo·(c)
For vectors x and y, let Mx = x(x'x)-lx' and My = y(y'y)-ly'. Show that MxMy = MyMx if and only if C(x) = C(y)or x 1. y.
Consider the matrix(a) Show that A is a projection matrix.(b) Is A a perpendicular projection matrix? Why or why not?(c) Describe the space that A projects onto and the space that A projects along.
Let A be an arbitrary projection matrix. Show that C(J - A) = C(A') 1. .Hints: Recall that C(A')l. is the null space of A. Show that (J - A) is a projection matrix.
Show that if A-is a generalized inverse of A, then so is for any choices of Bl and B2 with conformable dimensions.
Let A be positive definite with eigenvalues ),1, ... ,),n'Show that A-I has eigenvalues 1/),1."" 1/),n and the same eigenvectors as A.
Show that a noncentral chi-square distribution is stochastically larger than the central chi-square distribution with the same degrees of freedom. Show that a noncentral F distribution is
Let Y = (YI, Y2, Y3)' be a random vector. Suppose that E(Y) EM, where M is defined by M = {(a, a -b, 2b),la, bE R}.(a) Show that M is a vector space.(b) Find a basis for M.(c) Write a linear model
p, = (5,6,7), and[2 0 1] V= 0 3 2 .124 Find(a) the marginal distribution of YI,(b) the joint distribution of Y1 and Y2,(c) the conditional distribution of Y3 given Y1 = U1 and Y2 = U2,(d) the
The density of Y = (YI, Y2, Y3)' is where Q = 2y~ + y~ + Y5 + 2YIY2 - 8Yl - 4Y2 + 8.Find V-I and J1,.
an orthogonal matrix.(a) Find the distribution of O'Y.(b) Show that y. = (1/n)J'Y and that 82 = Y'OlO~Y/(n -1).(c) Show that y. and 82 are independent.Hint: Show that y'y = Y'OO'Y = Y'(1/n)JJ'Y +
that if Let Y = (YI,Y2)' have a N(O,J) distribution. Show A=[! ~]then the conditions of Theorem 1.3.7 implying independence of Y' AY and Y' BY are satisfied only if lal = 1/lbl and a = -b. What are

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