Chapter 14

Below are \(n=50\) computer-generated observations that are presumably a random sample from the exponential pdf, \(f_{Y}(y)=e^{-y}, y \geq 0\). Use Theorem 14.2.1 to test whether the difference between the sample median for these \(y_{i}\) 's \((=0.604)\) and the true median of \(f_{Y}(y)\) is statistically significant. Let \(\alpha=0.05\). \(\begin{array}{lllllll}0.27187 & 0.46495 & 0.19368 & 0.80433 & 1.25450 & 0.62962 & 1.88300 \\ 1.31951 & 2.53918 & 1.21187 & 0.95834 & 0.49017 & 0.87230 & 0.88571 \\ 1.41717 & 1.75994 & 0.60280 & 2.19654 & 0.00594 & 4.11127 & 0.24130 \\ 0.16473 & 0.08178 & 1.01424 & 0.60511 & 0.87973 & 0.06127 & 0.24758 \\ 0.54407 & 0.05267 & 0.75210 & 0.13538 & 0.42956 & 0.02261 & 1.20378 \\ 1.09271 & 1.88705 & 0.17500 & 0.50194 & 0.52122 & 0.02915 & 0.27348 \\ 0.08916 & 0.72997 & 0.37185 & 0.06500 & 1.47721 & 4.02733 & 0.64003 \\ 0.05603 & & & & & & \end{array}\)

Let \(Y_{1}, Y_{2}, \ldots, Y_{22}\) be a random sample of normally distributed random variables with an unknown mean \(\mu\) and a known variance of \(6.0\). We wish to test $$ \begin{gathered} H_{0}: \mu=10 \\ \text { versus } \\ H_{1}: \mu>10 \end{gathered} $$ Construct a large-sample sign test having a Type I error probability of \(0.05\). What will the power of the test be if \(\mu=11 ?\)

Suppose that \(n=7\) paired observations, \(\left(X_{i}, Y_{i}\right)\), are recorded, \(i=1,2, \ldots, 7\). Let \(p=P\left(Y_{i}>X_{i}\right)\). Write out the entire probability distribution for \(Y_{+}\), the number of positive differences among the set of \(Y_{i}-X_{i}\) 's, \(i=1,2, \ldots, 7\), assuming that \(p=\frac{1}{2} .\) What \(\alpha\) levels are possible for testing \(H_{0}: p=\frac{1}{2}\) versus \(H_{1}: p>\frac{1}{2}\) ?

In a marketing research test, twenty-eight adult males were asked to shave one side of their face with one brand of razor blade and the other side with a second brand. They were to use the blades for seven days and then decide which was giving the smoother shave. Suppose that nineteen of the subjects preferred blade A. Use a sign test to determine whether it can be claimed, at the \(0.05\) level, that the difference in preferences is statistically significant.

Two manufacturing processes are available for annealing a certain kind of copper tubing, the primary difference being in the temperature required. The critical response variable is the resulting tensile strength. To compare the methods, fifteen pieces of tubing were broken into pairs. One piece from each pair was randomly selected to be annealed at a moderate temperature, the other piece at a high temperature. The resulting tensile strengths (in tons/sq in.) are listed in the following table. Analyze these data with a Wilcoxon signed rank test. Use a two-sided alternative. Let \(\alpha=0.05\). \begin{tabular}{ccc} \hline \multicolumn{3}{c}{ Tensile Strengths (tons/sq in.) } \\ \hline \multicolumn{3}{c}{ Moderate } \\ Pair & Temperature & High \\ \hline 1 & \(16.5\) & \(16.9\) \\ 2 & \(17.6\) & \(17.2\) \\ 3 & \(16.9\) & \(17.0\) \\ 4 & \(15.8\) & \(16.1\) \\ 5 & \(18.4\) & \(18.2\) \\ 6 & \(17.5\) & \(17.7\) \\ 7 & \(17.6\) & \(17.9\) \\ 8 & \(16.1\) & \(16.0\) \\ 9 & \(16.8\) & \(17.3\) \\ 10 & \(15.8\) & \(16.1\) \\ 11 & \(16.8\) & \(16.5\) \\ 12 & \(17.3\) & \(17.6\) \\ 13 & \(18.1\) & \(18.4\) \\ 14 & \(17.9\) & \(17.2\) \\ 15 & \(16.4\) & \(16.5\) \\ \hline \end{tabular}

To measure the effect on coordination associated with mild intoxication, thirteen subjects were each given \(15.7 \mathrm{~mL}\) of ethyl alcohol per square meter of body surface area and asked to write a certain phrase as many times as they could in the space of one minute (127). The number of correctly written letters was then counted and scaled, with a scale value of 0 representing the score a subject not under the influence of alcohol would be expected to achieve. Negative scores indicate decreased writing speeds; positive scores, increased writing speeds. Use the signed rank test to determine whether the level of alcohol provided in this study had any effect on writing speed. Let \(\alpha=0.05 .\) Omit Subject 8 from your calculations. \begin{tabular}{crrr} \hline Subject & Score & Subject & Score \\ \hline 1 & \(-6\) & 8 & 0 \\ 2 & 10 & 9 & \(-7\) \\ 3 & 9 & 10 & 5 \\ 4 & \(-8\) & 11 & \(-9\) \\ 5 & \(-6\) & 12 & \(-10\) \\ 6 & \(-2\) & 13 & \(-2\) \\ 7 & 20 & & \\ \hline \end{tabular}

Suppose that the population being sampled is symmetric and we wish to test \(H_{0}: \tilde{\mu}=\tilde{\mu}_{0}\). Both the sign test and the signed rank test would be valid. Which procedure, if either, would you expect to have greater power? Why?

A sample of ten 40 -W light bulbs was taken from each of three manufacturing plants. The bulbs were burned until failure. The number of hours that each remained lit is listed in the following table. \begin{tabular}{rrr} \hline Plant I & Plant 2 & Plant 3 \\ \hline 905 & 1109 & 571 \\ 1018 & 1155 & 1346 \\ 905 & 835 & 292 \\ 886 & 1152 & 825 \\ 958 & 1036 & 676 \\ 1056 & 926 & 541 \\ 904 & 1029 & 818 \\ 856 & 1040 & 90 \\ 1070 & 959 & 2246 \\ 1006 & 996 & 104 \\ \hline \end{tabular} (a) Test the hypothesis that the median lives of bulbs produced at the three plants are all the same. Use the \(0.05\) level of significance. (b) Are the mean lives of bulbs produced at the three plants all the same? Use the analysis of variance with \(\alpha=0.05\). (c) Change the observation " 2246 " in the third column to "1500" and redo part (a). How does this change affect the hypothesis test? (d) Change the observation " 2246 " in the third column to "1500" and redo part (b). How does this change affect the hypothesis test?

Until its recent indictment as a possible carcinogen, cyclamate was a widely used sweetener in soft drinks. The following data show a comparison of three laboratory methods for determining the percentage of sodium cyclamate in commercially produced orange drink. All three procedures were applied to each of twelve samples (165). Percent Sodium Cyclamate (w/w) \begin{tabular}{cccc} \hline & \multicolumn{3}{c}{ Method } & \\ \cline { 2 - 4 } Sample & Picryl Chloride & Davies & AOAC \\ \hline 1 & \(0.598\) & \(0.628\) & \(0.632\) \\ 2 & \(0.614\) & \(0.628\) & \(0.630\) \\ 3 & \(0.600\) & \(0.600\) & \(0.622\) \\ 4 & \(0.580\) & \(0.612\) & \(0.584\) \\ 5 & \(0.596\) & \(0.600\) & \(0.650\) \\ 6 & \(0.592\) & \(0.628\) & \(0.606\) \\ 7 & \(0.616\) & \(0.628\) & \(0.644\) \\ 8 & \(0.614\) & \(0.644\) & \(0.644\) \\ 9 & \(0.604\) & \(0.644\) & \(0.624\) \\ 10 & \(0.608\) & \(0.612\) & \(0.619\) \\ 11 & \(0.602\) & \(0.628\) & \(0.632\) \\ 12 & \(0.614\) & \(0.644\) & \(0.616\) \\ \hline \end{tabular} Use Friedman's test to determine whether the differences from method to method are statistically significant. Let \(\alpha=0.05\).

Suppose that \(k\) treatments are to be applied within each of \(b\) blocks. Let \(\bar{r}_{\text {.. denote the average of the }} b k\) ranks and let \(\bar{r}_{. j}=(1 / b) r_{. j} .\) Show that the Friedman statistic given in Theorem 14.5.1 can also be written $$ g=\frac{12 b}{k(k+1)} \sum_{j=1}^{k}\left(\bar{r}_{. j}-\bar{r}_{. .}\right)^{2} $$ What analysis of variance expression does this resemble?

The data in the table examine the relationship between stock market changes (1) during the first few days in January and (2) over the course of the entire year. Included are the years from 1950 through \(1986 .\) (a) Use Theorem 14.6.1 to test the randomness of the January changes (relative to the number of runs up and down). Let \(\alpha=0.05\). (b) Use Theorem 14.6.1 to test the randomness of the annual changes. Let \(\alpha=0.05\). | | % Change for | | | :--- | :---: | :--- | | | First 5 Days in | % Change | | Year | Jan., x | for Year, y | | 1950 | 2.0 | 21.8 | | :--- | ---: | ---: | | 1951 | 2.3 | 16.5 | | 1952 | 0.6 | 11.8 | | 1953 | -0.9 | -6.6 | | 1954 | 0.5 | 45.0 | | 1955 | -1.8 | 26.4 | | 1956 | -2.1 | 2.6 | | 1957 | -0.9 | -14.3 | | 1958 | 2.5 | 38.1 | | 1959 | 0.3 | 8.5 | | 1960 | -0.7 | -3.0 | | 1961 | 1.2 | 23.1 | | 1962 | -3.4 | -11.8 | | 1963 | 2.6 | 18.9 | | 1964 | 1.3 | 13.0 | | 1965 | 0.7 | 9.1 | | 1966 | 0.8 | -13.1 | | 1967 | 3.1 | 20.1 | | 1968 | 0.2 | 7.7 | | 1969 | -2.9 | -11.4 | | ---: | ---: | ---: | | 1970 | 0.7 | 0.1 | | 1971 | 0.0 | 10.8 | | 1972 | 1.4 | 15.6 | | 1973 | 1.5 | -17.4 | | 1974 | -1.5 | -29.7 | | 1975 | 2.2 | 31.5 | | 1976 | 4.9 | 19.1 | | 1977 | -2.3 | -11.5 | | 1978 | -4.6 | 1.1 | | 1979 | 2.8 | 12.3 | | 1980 | 0.9 | 25.8 | | 1981 | -2.0 | -9.7 | | 1982 | -2.4 | 14.8 | | 1983 | 3.2 | 17.3 | | 1984 | 2.4 | 1.4 | | 1985 | -1.9 | 26.3 | | 1986 | -1.6 | 14.6 |

Listed below for two consecutive fiscal years are the monthly numbers of passenger boardings at a Florida airport. Use Theorem 14.6.1 to test whether these | Month | Passenger Boardings | Month | Passenger Boardings | | :--- | :---: | :--- | :---: | | July | 41,388 | July | 44,148 | | Aug. | 44,880 | Aug. | 42,038 | | Sept. | 33,556 | Sept. | 35,157 | | Oct. | 34,805 | Oct. | 39,568 | | Nov. | 33,025 | Nov. | 34,185 | | Dec. | 34,873 | Dec. | 37,604 | | Jan. | 31,330 | Jan. | 28,231 | | Feb. | 30,954 | Feb. | 29,109 | | March | 32,402 | March | 38,080 | | April | 38,020 | April | 34,184 | | May | 42,828 | May | 39,842 | | June | 41,204 | June | 46,727 | twenty-four observations can be considered a random sequence, relative to the number of runs up and down. Let \(\alpha=0.05\).

On the next page is a partial statistical summary of the first twenty-four Super Bowls (36). Of particular interest to advertisers is the network share that each game garnered. Can those shares be considered a random sequence, relative to the number of runs up and down? Test the appropriate hypothesis at the \(\alpha=0.05\) level of significance. | | | | | Network | | :---: | :---: | :---: | :---: | :---: | | Game, | Winner, | | MVP Is | Share | | Year | Loser | Score | QB | (network) | | I | Green Bay (NFL) | 35 | I | 79 | | :--- | :--- | ---: | :--- | :--- | | 1967 | Kansas City (AFL) | 10 | | (CBS/NBC | | | | | | combined) | | II | Green Bay (NFL) | 33 | I | 68 | | 1968 | Oakland (AFL) | 14 | | (CBS) | | III | NY Jets (AFL) | 16 | I | 7 I | | I969 | Baltimore (NFL) | 7 | | (NBC) | | IV | Kansas City (AFL) | 23 | I | 69 | | 1970 | Minnesota (NFL) | 7 | | (CBS) | | V | Baltimore (AFC) | 16 | 0 | 75 | | 197I | Dallas (NFC) | 13 | | (NBC) | | VI | Dallas (NFC) | 24 | 1 | 74 | | 1972 | Miami (AFC) | 3 | | (CBS) | | VII | Miami (AFC) | 14 | 0 | 72 | | 1973 | Washington (NFC) | 7 | | (NBC) | | VIII | Miami (AFC) | 24 | 0 | 73 | | 1974 | Minnesota (NFC) | 7 | | (CBS) | | IX | Pittsburgh (AFC) | 16 | 0 | 72 | | 1975 | Minnesota (NFC) | 6 | | (NBC) | | X | Pittsburgh (AFC) | 21 | 0 | 78 | | 1976 | Dallas (NFC) | 17 | | (CBS) | | :--- | :--- | ---: | :--- | :--- | | XI | Oakland (AFC) | 32 | 0 | 73 | | 1977 | Minnesota (NFC) | 14 | | (NBC) | | XII | Dallas (NFC) | 27 | 0 | 67 | | 1978 | Denver (AFC) | 10 | | (CBS) | | XIII | Pittsburgh (AFC) | 35 | I | 74 | | 1979 | Dallas (NFC) | 31 | | (NBC) | | XIV | Pittsburgh (AFC) | 31 | I | 67 | | 1980 | Los Angeles (AFC) | 19 | | (CBS) | | XV | Oakland (AFC) | 27 | I | 63 | | 198I | Philadelphia (NFC) | 10 | | (NBC) | | XVI | San Francisco (NFC) | 26 | I | 73 | | 1982 | Cincinnati (AFC) | 21 | | (CBS) | | XVII | Washington (NFC) | 27 | 0 | 69 | | 1983 | Miami (AFC) | 17 | | (NBC) | | XVIII | LA Raiders (AFC) | 38 | 0 | 7 I | | 1984 | Washington (NFC) | 9 | | (CBS) | | XIX | San Francisco (NFC) | 38 | I | 63 | | 1985 | Miami (AFC) | 16 | | (ABC) | | XX | Chicago (NFC) | 46 | 0 | 70 | | 1986 | New England (AFC) | 10 | | (NBC) | | XXI | NY Giants (NFC) | 39 | 1 | 66 | | 1987 | Denver (AFC) | 20 | | (CBS) | | XXII | Washington (NFC) | 42 | 1 | 62 | | 1988 | Denver (AFC) | 10 | | (ABC) | | XXIII | San Francisco (NFC) | 20 | 0 | 68 | | | | | | Network | | :--- | :---: | :---: | :---: | :---: | | Game, Year | Winner, Loser | Score | MVP Is QB | Share (network) | | 1989 | Cincinnati (AFC) | 16 | | (NBC) | | XXIV | San Francisco (NFC) | 55 | ? | 63 | | 1990 | Denver (AFC) | 10 | | (CBS) |

Below are the lengths (in \(\mathrm{mm}\) ) of furniture dowels recorded as part of an ongoing quality-control program. Listed are the measurements made on thirty samples (each of size 4) taken in order from the assembly line. Is the variation in the sample averages random with respect to the number of runs up and down? Do an appropriate hypothesis test at the \(\alpha=0.05\) level of significance. | Sample | y_(1) | y_(2) | y_(3) | y_(4) | bar(y) | | :---: | :---: | :---: | :---: | :---: | :---: | | 1 | 46.1 | 44.4 | 45.3 | 44.2 | 45.0 | | 2 | 46.0 | 45.4 | 42.5 | 44.4 | 44.6 | | 3 | 44.3 | 44.0 | 45.4 | 43.9 | 44.4 | | 4 | 44.9 | 43.7 | 45.2 | 44.8 | 44.7 | | 5 | 43.0 | 45.3 | 45.9 | 43.8 | 44.5 | | 6 | 46.0 | 43.2 | 44.4 | 43.7 | 44.3 | | 7 | 46.0 | 44.6 | 45.4 | 46.4 | 45.6 | | 8 | 46.1 | 45.5 | 45.0 | 45.5 | 45.5 | | 9 | 42.8 | 45.1 | 44.9 | 44.3 | 44.3 | | 10 | 45.0 | 46.7 | 43.0 | 44.8 | 44.9 | | 11 | 45.5 | 44.5 | 45.1 | 47.1 | 45.6 | | 12 | 45.8 | 44.6 | 44.8 | 45.1 | 45.1 | | 13 | 45.1 | 45.4 | 46.0 | 45.4 | 45.5 | | 14 | 44.6 | 43.8 | 44.2 | 43.9 | 44.1 | | 15 | 44.8 | 45.5 | 45.2 | 46.2 | 45.4 | | 16 | 45.8 | 44.1 | 43.3 | 45.8 | 44.8 | | 17 | 44.1 | 44.8 | 46.1 | 45.5 | 45.1 | | 18 | 44.5 | 43.6 | 45.1 | 46.9 | 45.0 | | 19 | 45.2 | 43.1 | 46.3 | 46.4 | 45.3 | | 20 | 45.9 | 46.8 | 46.8 | 45.8 | 46.3 | | 21 | 44.0 | 44.7 | 46.2 | 45.4 | 45.1 | | 22 | 43.4 | 44.6 | 45.4 | 44.4 | 44.5 | | 23 | 43.1 | 44.6 | 44.5 | 45.8 | 44.5 | | 24 | 46.6 | 43.3 | 45.1 | 44.2 | 44.8 | | 25 | 46.2 | 44.9 | 45.3 | 46.0 | 45.6 | | 26 | 42.5 | 43.4 | 44.3 | 42.7 | 43.2 | | 27 | 43.4 | 43.3 | 43.4 | 43.5 | 43.4 | | 28 | 42.3 | 42.4 | 46.6 | 42.3 | 43.4 | | 29 | 41.9 | 42.9 | 42.0 | 42.9 | 42.4 | | 30 | 43.2 | 43.5 | 42.2 | 44.7 | 43.4 |

Listed below are forty ordered computergenerated observations that presumably represent a normal distribution with \(\mu=5\) and \(\sigma=2\). Can the sample be considered random with respect to the number of runs up and down? | Obs. # | y_(i) | Obs. # | y_(i) | Obs. # | y_(i) | Obs. # | y_(i) | | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | | I | 7.0680 | 11 | 7.6979 | 21 | 5.9828 | 31 | 5.2625 | | 2 | 4.0540 | 12 | 4.4338 | 22 | 1.4614 | 32 | 5.9047 | | 3 | 6.6165 | 13 | 5.6538 | 23 | 9.2655 | 33 | 4.6342 | | 4 | 1.2166 | 14 | 8.0791 | 24 | 4.9281 | 34 | 5.3089 | | 5 | 4.6158 | 15 | 4.7458 | 25 | 10.5561 | 35 | 5.4942 | | 6 | 7.7540 | 16 | 3.5044 | 26 | 6.1738 | 36 | 6.6914 | | 7 | 7.7300 | 17 | 1.3071 | 27 | 5.4895 | 37 | 1.4380 | | 8 | 6.5109 | 18 | 5.7893 | 28 | 3.6629 | 38 | 8.2604 | | 9 | 3.8933 | 19 | 4.5241 | 29 | 3.7223 | 39 | 5.0209 | | 10 | 2.7533 | 20 | 5.3291 | 30 | 3.5211 | 40 | 0.5544 |