The article "Can We Really Walk Straight?" (Amer. J. Phys. Anthropol., 1992: 19-27) reported on an experiment in which each of 20 healthy men was asked to walk as straight as possible to a target $60 {m}$ away at normal speed. Consider the following observations on cadence (number of strides per second):

$$\begin{array}{llllllllll}.95 & .85 & .92 & .95 & .93 & .86 & 1.00 & .92 & .85 & .81\end{array}$$

$$\begin{array}{llllllllll}.78 & .93 & .93 & 1.05 & .93 & 1.06 & 1.06 & .96 & .81 & .96\end{array}$$

Use the methods developed in this chapter to summarize the data; include an interpretation or discussion wherever appropriate.

The brands of the last 45 digital single-lens reflex (SLR) cameras sold on a popular Internet auction site are listed here. Summarize the distribution of camera brands with a frequency table and a relative frequency table.

$$\begin{array}{l} \text{Canon} & \text{Sony} & \text{Canon} & \text{Nikon} & \text{Fujifilm}\\ \text{Nikon} & \text{Canon} & \text{Sony} & \text{Canon} & \text{Canon}\\ \text{Nikon} & \text{Canon} & \text{Nikon} & \text{Canon} & \text{Canon}\\ \text{Canon} & \text{Nikon} & \text{Fujifilm} & \text{Canon} & \text{Nikon}\\ \text{Nikon} & \text{Canon} & \text{Canon} & \text{Canon} & \text{Canon}\\ \text{Olympus} & \text{Canon} & \text{Canon} & \text{Canon} & \text{Nikon}\\ \text{Olympus} & \text{Sony} & \text{Canon} & \text{Canon} & \text{Sony}\\ \text{Canon} & \text{Nikon} & \text{Sony} & \text{Canon} & \text{Fujifilm}\\ \text{Nikon} & \text{Canon} & \text{Nikon} & \text{Canon} & \text{Sony}\\ \end{array}$$

A mutual fund is a professionally managed investment scheme that pools money from many investors and invests in a variety of securities. Growth funds focus primarily on increasing the value of investments, whereas blended funds seek a balance between current income and growth. Here is data on the expense ratio (expenses as a % of assets) for samples of 20 large-cap balanced funds and 20 large-cap growth funds (“largecap” refers to the sizes of companies in which the funds invest; the population sizes are 825 and 762, respectively): $\textbf{Bl : }$1.03, 1.23, 1.10, 1.64, 1.30, 1.27, 1.25, 0.78, 1.05$\quad\quad$0.64, 0.94, 2.86, 1.05, 0.75, 0.09, 0.79, 1.61, 1.26 $\quad\quad$0.93, 0.84$\textbf{Gr : }$0.52, 1.06, 1.26, 2.17, 1.55, 0.99, 1.10, 1.07, 1.81$\quad\quad$2.05, 0.91, 0.79, 1.39, 0.62, 1.52, 1.02, 1.10, 1.78 $\quad\quad$1.01, 1.15 (a.) Calculate and compare the values of $\overline{x}\text{, }\tilde{x}$ and s for the two types of funds. (b.) Construct a comparative boxplot for the two types of funds, and comment on interesting features.

The Apprentice contestants' performance ratings. Refer to the Significance (April 2015) study of contestants' performances on the United Kingdom's version of the TV show, The Apprentice, Exercise $2.9$ (p. 49). Recall that the performance of each 159 contestants was rated on a 20-point scale. Contestants were also divided into two groups: those who played for a job and those who played for a business partnership. These data (simulated, based on statistics reported in the article) are saved in the accompanying file. Descriptive statistics for each of the two groups of contestants are displayed in the accompanying Minitab printout. Determine whether the performance ratings of contestants who played for a job are approximately normally distributed.

In 1997 a woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard (Genessy v. Digital Equipment Corp.). The injury awarded about $3.5 million for pain and suffering, but the court then set aside that award as being unreasonable compensation. In making this determination, the court identified a “normative” group of 27 similar cases and specified a reasonable award as one within two standard deviations of the mean of the awards in the 27 cases. The 27 awards were (in$1000s)

$$37\text{, }60\text{, }75\text{, }115\text{, }135\text{, }140\text{, }149\text{, }150\text{, }238\text{, }290$$

$$340\text{, }410\text{, }600\text{, }750\text{, }750\text{, }750\text{, }1050\text{, }1100\text{, }1139$$

$$1150\text{, }1200\text{, }1200\text{, }1250\text{, }1576\text{, }1700\text{, }1825\text{, }2000$$

From the data : $\sum x_{i} = 20,179$ and $\sum x_{i}^{2} = 24,657,511$ What is the maximum possible amount that could be awarded under the two- standard deviation rule?

Following is the data on endotoxin concentration in settled dust both for a sample of urban homes and for a sample of farm homes:

$$\text{U: }6.0 \text{, }5.0 \text{, }11.0 \text{, }33.0 \text{, }4.0 \text{, }5.0 \text{, }80.0 \text{, }18.0 \text{, }35.0$$

$$\quad\text{ } 17.0 \text{, }23.0$$

$$\text{F: }4.0 \text{, }14.0 \text{, }11.0 \text{, }9.0 \text{, }9.0 \text{, }8.0 \text{, }4.0 \text{, }20.0 \text{, }5.0 \text{, }8.9$$

$$\quad\text{ } 21.0 \text{, }9.2 \text{, }3.0 \text{, }2.0 \text{, }0.3$$

(a.) Determine the value of the sample standard deviation for each sample, interpret these values, and then contrast variability in the two samples. [Hint: sum of terms is 237.0 and 128.4 for the urban sample and the farm sample respectively, and sum of square of terms is 10079 and 1617.94 for the urban sample and the farm sample respectively.] (b.) Compute the fourth spread for each sample and compare. Do the fourth spreads convey the same message about variability that the standard deviations do? Explain. (c.) Endotoxin concentrations in dust bag dust are also provided below :

$$\text{U: }34.0 \text{, }49.0 \text{, }13.0 \text{, }33.0 \text{, }24.0 \text{, }24.0 \text{, }35.0 \text{, }104.0$$

$$\quad\text{ } 34.0 \text{, }40.0 \text{, }38.0 \text{, }1.0$$

$$\text{F: }2.0 \text{, }64.0 \text{, }6.0 \text{, }17.0 \text{, }35.0 \text{, }11.0 \text{, }17.0 \text{, }13.0 \text{, }5.0$$

$$\quad\text{ } 27.0 \text{, }23.0 \text{, }28.0 \text{, }10.0 \text{, }13.0 \text{, }0.2$$

Construct a comparative boxplot and compare and contrast the four samples.

The accompanying observations on stabilized viscosity (cP) for specimens of a certain grade of asphalt with 18% rubber added are from the article “Viscosity Characteristics of Rubber-Modified Asphalts” (J. of Materials in Civil Engr., 1996: 153-156): 2781 2900 3013 2856 2888 a. What are the values of the sample mean and sample median? b. Calculate the sample variance using the computational formula. [Hint: First subtract a convenient number from each observation.]

Performing a Sign Test

a) write the claim mathematically and identify $H_0$ and $H_w$

b) determine the critical value,

c) calculate the test statistic,

d) decide whether to reject or fail to reject the null hypothesis

e) interpret the decision in the context of the original claim.

A financial services accountant estimates that the median amount of financial debt for families holding such debts is less than $50.000. In a random sample of 70 families holding debt, the debt for 24 families was less than$ 50,000 and the debt for 46 families was greater than $50,000. At$\alpha=0.025$, can you support the accountant's estimate?

(a.) If a constant $c$ is added to each $x_{i}$ in a sample, yielding $y_{i} = x_{i}+c$ , how do the sample mean and median of the $y_{i}$s relate to the mean and median of the $x_{i}$s? Verify your conjectures. (b.) If each $x_{i}$ is multiplied by a constant c, yielding $y_{i}=cx_{i}$. Then answer the question of part (a). Again, verify your conjectures.

Compute the sample median, 25% trimmed mean, 10% trimmed mean, and sample mean for the lifetime data given below, and compare these measures.

$$11 \text{, }14 \text{, }20 \text{, }23 \text{, }31 \text{, }36 \text{, }39 \text{, }44 \text{, }47 \text{, }50$$

$$59 \text{, }61 \text{, }65 \text{, }67 \text{, }68 \text{, }71 \text{, }74 \text{, }76 \text{, }78 \text{, }79$$

$$81 \text{, }84 \text{, }85 \text{, }89 \text{, }91 \text{, }93 \text{, }96 \text{, }99 \text{, }101 \text{, }104$$

$$105 \text{, }105 \text{, }112 \text{, }118 \text{, }123 \text{, }136 \text{, }139 \text{, }141 \text{, }148 \text{, }158$$

$$161 \text{, }168 \text{, }184 \text{, }206 \text{, }248 \text{, }263 \text{, }289 \text{, }322 \text{, }388 \text{, }513$$

The article “Snow Cover and Temperature Relationships in North America and Eurasia” (J. Climate and Applied Meteorology, 1983: 460-469) used statistical techniques to relate the amount of snow cover on each continent to average continental temperature. Data presented there included the following ten observations on October snow cover for Eurasia during the years 1970-1979 (in million $\mathrm{km}^{2}$):

$$\begin{array}{llllllllll} 6.5 & 12.0 & 14.9 & 10.0 & 10.7 & 7.9 & 21.9 & 12.5 & 14.5 & 9.2 \end{array}$$

What would you report as a representative, or typical, value of October snow cover for this period, and what prompted your choice?

The minimum injection pressure (psi) for injection molding specimens of high amylose corn was determined for eight different specimens (higher pressure corresponds to greater processing difficulty), resulting in the following observations (from “Thermoplastic Starch Blends with a Polyethylene-Co-Vinyl Alcohol: Processability and Physical Properties,” Polymer Engr. and Science, 1994: 17-23):

$$\begin{matrix} \text{15.0} & \text{13.0} & \text{18.0} & \text{14.5} & \text{12.0} & \text{11.0} & \text{8.9} & \text{8.0}\\ \end{matrix}$$

a. Determine the values of the sample mean, sample median, and 12.5% trimmed mean, and compare these values. b. By how much could the smallest sample observation, currently 8.0, be increased without affecting the value of the sample median? c. Suppose we want the values of the sample mean and median when the observations are expressed in kilograms per square inch (ksi) rather than psi. Is it necessary to reexpress each observation in ksi, or can the values calculated in part (a) be used directly? [Hint: 1 kg=2.2 lb.]

Human measurements provide a rich area of application for statistical methods. The article “A Longitudinal Study of the Development of Elementary School Children’s Private Speech” (Merrill-Palmer Q., 1990: 443-63) reported on a study of children talking to themselves (private speech). It was thought that private speech would be related to IQ, because IQ is supposed to measure mental maturity, and it was known that private speech decreases as students progress through the primary grades. The study included 33 students whose first-grade IQ scores are given here:

$$\begin{matrix} \text{82} & \text{96} & \text{99} & \text{102} & \text{103} & \text{103} & \text{106} & \text{107} & \text{108} & \text{108} & \text{108} & \text{108}\\ \text{109} & \text{110} & \text{110} & \text{111} & \text{113} & \text{113} & \text{113} & \text{113} & \text{115} & \text{115} & \text{118} & \text{118}\\ \text{119} & \text{121} & \text{122} & \text{122} & \text{127} & \text{132} & \text{136} & \text{140} & \text{146}\\ \end{matrix}$$

Describe the data and comment on any interesting features.

The amount of flow through a solenoid valve in an automobile’s pollution-control system is an important characteristic. An experiment was carried out to study how flow rate depended on three factors: armature length, spring load, and bobbin depth. Two different levels (low and high) of each factor were chosen, and a single observation on flow was made for each combination of levels. (a.) The resulting data set consisted of how many observations? (b.) Is this an enumerative or analytic study? Explain your reasoning.