91影视

a. Give three different examples of concrete populations and three different examples of hypothetical populations.

b. For one each of your concrete and your hypothetical populations, give an example of a probability question and an example of an inferential statistics question.

For each of the following hypothetical populations, give

a plausible sample of size 4:

a. All distances that might result when you throw a football

b. Page lengths of books published 5 years from now

c. All possible earthquake-strength measurements (Richter scale) that might be recorded in California during the next year

d. All possible yields (in grams) from a certain chemical reaction carried out in a laboratory.

A trial has just resulted in a hung jury because eight members of the jury were in favour of a guilty verdict and the other four were for acquittal. If the jurors leave the jury room in random order and each of the first four leaving the room is accosted by a reporter in quest of an interview, what is the\({\rm{pmf}}\)of\({\rm{X = }}\)the number of jurors favouring acquittal among those interviewed? How many of those favouring acquittal do you expect to be interviewed?

Using a long rod that has length \({\rm{\mu }}\)you are going to lay out a square plot in which the length of each side is \({\rm{\mu }}\).Thus the area of the plot will be \({{\rm{\mu }}^{\rm{2}}}\)However, you do not know the value of \({\rm{\mu }}\), so you decide to make \({\rm{n}}\)independent measurements \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}\)of the length. Assume that each \({{\rm{X}}_{\rm{i}}}\)has mean \({\rm{\mu }}\) (unbiased measurements) and variance \({{\rm{\sigma }}^{\rm{2}}}\).

a. Show that \({{\rm{\bar X}}^{\rm{2}}}\)is not an unbiased estimator for \({{\rm{\mu }}^{\rm{2}}}\). (Hint: For any \({\rm{Y,E}}\left( {{{\rm{Y}}^{\rm{2}}}} \right){\rm{ = V(Y) + (E(Y)}}{{\rm{)}}^{\rm{2}}}\)Apply this with \({\rm{Y = \bar X}}\))

b. For what value of \({\rm{k}}\)is the estimator \({{\rm{\bar X}}^{\rm{2}}}{\rm{ - k}}{{\rm{S}}^{\rm{2}}}\)unbiased for \({{\rm{\mu }}^{\rm{2}}}\)? (Hint: Compute \({\rm{E}}\left( {{{{\rm{\bar X}}}^{\rm{2}}}{\rm{ - k\;}}{{\rm{S}}^{\rm{2}}}} \right)\))

Consider the strength data for beams given in Example

1.2.

a. Construct a stem-and-leaf display of the data. What appears to be a representative strength value? Do the observations appear to be highly

concentrated about the representative value or rather spread out?

b. Does the display appear to be reasonably symmetric about a representative value, or would you describe its shape in some other way?

c. Do there appear to be any outlying strength values?

d. What proportion of strength observations in this sample exceeds 10 MPa?

Let \({\rm{X}}\) denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is

\({\rm{F(x) = }}\left\{ {\begin{array}{*{20}{l}}{\rm{0}}&{{\rm{x < 0}}}\\{\frac{{{{\rm{x}}^{\rm{2}}}}}{{\rm{4}}}}&{{\rm{0拢 x < 2}}}\\{\rm{1}}&{{\rm{2\拢 x}}}\end{array}} \right.\)

a. Calculate\({\rm{P(X拢 1)}}\).

b. Calculate\({\rm{P(}}{\rm{.5拢 X拢 1)}}\).

c. Calculate\({\rm{P(X > 1}}{\rm{.5)}}\).

d. What is the median checkout duration \({\rm{\tilde \mu }}\) ? (solve\({\rm{5 = F(\tilde \mu ))}}\).

e. Obtain the density function\({\rm{f(x)}}\).

f. Calculate\({\rm{E(X)}}\).

g. Calculate \({\rm{V(X)}}\)and\({{\rm{\sigma }}_{\rm{X}}}\).

h. If the borrower is charged an amount \({\rm{h(X) = }}{{\rm{X}}^{\rm{2}}}\) when checkout duration is\({\rm{X}}\), compute the expected charge\({\rm{E(h(X))}}\).

The accompanying specific gravity values for various wood types used in construction appeared in the article 鈥淏olted Connection Design Values Based on European Yield Model鈥� (J. of Structural Engr., 1993: 2169鈥�2186):

Construct a stem-and-leaf display using repeated stems, and comment on any interesting features of the display.

The cdf for \({\rm{X( = measurement error)}}\) of Exercise \({\rm{3}}\) is

\({\rm{F(x) = }}\left\{ {\begin{array}{*{20}{c}}{\rm{0}}&{{\rm{x < - 2}}}\\{\frac{{\rm{1}}}{{\rm{2}}}{\rm{ + }}\frac{{\rm{3}}}{{{\rm{32}}}}\left( {{\rm{4x - }}\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right)}&{{\rm{ - 2}} \le {\rm{x < 2}}}\\{\rm{1}}&{{\rm{2}} \le {\rm{x}}}\end{array}} \right.\)

The accompanying summary data on CeO2 particlesizes (nm) under certain experimental conditions wasread from a graph in the article 鈥淣anoceria鈥擡nergetics of Surfaces, Interfaces and WaterAdsorption鈥� (J. of the Amer. Ceramic Soc., 2011:3992鈥�3999):

3.0鈭�<3.5 3.5鈭�<4.0 4.0鈭�<4.5 4.5鈭�<5.0 5.0鈭�<5.5

5 15 27 34 22

5.5鈭�<6.0 6.0鈭�<6.5 6.5鈭�<7.0 7.0鈭�<7.5 7.5鈭�<8.0

14 7 2 4 1

a. What proportion of the observations are less than 5?

b. What proportion of the observations are at least 6?

c. Construct a histogram with relative frequency on the vertical axis and comment on interesting features. In particular, does the distribution of particle sizes appear to be reasonably symmetric or somewhat skewed? (Note:The investigators fit lognormaldistribution to the data; this is discussed in Chapter 4.)

d. Construct a histogram with density on the vertical axis and compare to the histogram in (c).

Allowable mechanical properties for structural design of metallic aerospace vehicles requires an approved method for statistically analyzing empirical test data. The article 鈥淓stablishing Mechanical Property Allowables for Metals鈥� (J. of Testing and Evaluation, 1998: 293鈥�299) used the accompanying data on tensile ultimate strength (ksi) as a basis for addressing the difficulties in developing such a method.

122.2 124.2 124.3 125.6 126.3 126.5 126.5 127.2 127.3

127.5 127.9 128.6 128.8 129.0 129.2 129.4 129.6 130.2

130.4 130.8 131.3 131.4 131.4 131.5 131.6 131.6 131.8

131.8 132.3 132.4 132.4 132.5 132.5 132.5 132.5 132.6

132.7 132.9 133.0 133.1 133.1 133.1 133.1 133.2 133.2

133.2 133.3 133.3 133.5 133.5 133.5 133.8 133.9 134.0

134.0 134.0 134.0 134.1 134.2 134.3 134.4 134.4 134.6

134.7 134.7 134.7 134.8 134.8 134.8 134.9 134.9 135.2

135.2 135.2 135.3 135.3 135.4 135.5 135.5 135.6 135.6

135.7 135.8 135.8 135.8 135.8 135.8 135.9 135.9 135.9

135.9 136.0 136.0 136.1 136.2 136.2 136.3 136.4 136.4

136.6 136.8 136.9 136.9 137.0 137.1 137.2 137.6 137.6

137.8 137.8 137.8 137.9 137.9 138.2 138.2 138.3 138.3

138.4 138.4 138.4 138.5 138.5 138.6 138.7 138.7 139.0

139.1 139.5 139.6 139.8 139.8 140.0 140.0 140.7 140.7

140.9 140.9 141.2 141.4 141.5 141.6 142.9 143.4 143.5

143.6 143.8 143.8 143.9 144.1 144.5 144.5 147.7 147.7

a. Construct a stem-and-leaf display of the data by first deleting (truncating) the tenths digit and then repeating each stem value five times (once for leaves 1 and 2, a second time for leaves 3 and 4, etc.). Why is it relatively easy to identify a representative strength value?

b. Construct a histogram using equal-width classes with the first class having a lower limit of 122 and an upper limit of 124. Then comment on any interesting features of the histogram.