91Ó°ÊÓ

Data on tipping percent for 20 restaurant tables, consistent with summary statistics given in the paper "Beauty and the Labor Market: Evidence from Restaurant Servers" (unpublished manuscript by Matt Parrett, 2007 ), are: $$ \begin{array}{rrrrrrrr} 0.0 & 5.0 & 45.0 & 32.8 & 13.9 & 10.4 & 55.2 & 50.0 \\ 10.0 & 14.6 & 38.4 & 23.0 & 27.9 & 27.9 & 105.0 & 19.0 \\ 10.0 & 32.1 & 11.1 & 15.0 & & & & \end{array} $$

Increasing joint extension is one goal of athletic trainers. In a study to investigate the effect of therapy that uses ultrasound and stretching (Trae Tashiro, Masters Thesis, University of Virginia, 2004 ), passive knee extension was measured after treatment. Passive knee extension (in degrees) is given for each of 10 study participants. $$ \begin{array}{llllllllll} 59 & 46 & 64 & 49 & 56 & 70 & 45 & 52 & 63 & 52 \end{array} $$ Which would you choose to describe center and variability the mean and standard deviation or the median and interquartile range? Justify your choice.

The accompanying data are a subset of data read from a graph in the paper "Ladies First? A Field Study of Discrimination in Coffee Shops" (Applied Economics [April, 2008] . The data are the waiting times (in seconds) between ordering and receiving coffee for 19 female customers at a Boston coffee shop. $$ \begin{aligned} &\begin{array}{llllllll} 60 & 80 & 80 & 100 & 100 & 100 & 120 & 120 \end{array}\\\ &\begin{array}{llllllll} 120 & 140 & 140 & 150 & 160 & 180 & 200 & 200 \end{array}\\\ &\begin{array}{lll} 220 & 240 & 380 \end{array} \end{aligned} $$ a. Calculate the mean and standard deviation for this data set. b. Delete the observation of 380 and recalculate the mean and standard deviation. How do these values compare to the values calculated in Part (a)? What does this suggest about using the mean and standard deviation as measures of center and variability for a data set with outliers?

The article "Caffeinated Energy Drinks-A Growing Problem" (Drug and Alcohol Dependence [2009]: 1-10) gave the accompanying data (on the next page) on caffeine concentration (mg/ounce) for eight top-selling energy drinks. a. What is the mean caffeine concentration for this set of energy drinks? b. Coca-Cola has 2.9 mg/ounce of caffeine and Pepsi Cola has \(3.2 \mathrm{mg} /\) ounce of caffeine. Write a sentence explaining how the caffeine concentration of top-selling energy drinks compares to that of these colas. $$ \begin{array}{|lc|} \hline \text { Energy Drink } & \begin{array}{c} \text { Caffeine Concentration } \\ \text { (mg/ounce) } \end{array} \\ \hline \text { Red Bull } & 9.6 \\ \text { Monster } & 10.0 \\ \text { Rockstar } & 10.0 \\ \text { Full Throttle } & 9.0 \\ \text { No Fear } & 10.9 \\ \text { Amp } & 8.9 \\ \text { SoBe Adrenaline Rush } & 9.5 \\ \text { Tab Energy } & 9.1 \\ \hline \end{array} $$

Acrylamide, a possible cancer-causing substance, forms in high-carbohydrate foods cooked at high temperatures. Acrylamide levels can vary widely even within the same type of food. An article appearing in the journal Food Chemistry (March 2014, 204-211) included the following acrylamide content (in nanograms/gram) for five brands of bisquits: $$ \begin{array}{lllll} 345 & 292 & 334 & 276 & 248 \end{array} $$ a. Calculate the mean acrylamide level. For each data value, calculate the deviation from the mean. b. Verify that, except for the effect of rounding, the sum of the five deviations from the mean is equal to 0 for this data set. (If you rounded the sample mean or the deviations, your sum may not be exactly zero, but it should still be close to zero.) c. Use the deviations from Part (a) to calculate the variance and standard deviation for this data set.

Although bats are not known for their eyesight, they are able to locate prey (mainly insects) by emitting high-pitched sounds and listening for echoes. A paper appearing in Animal Behaviour ("The Echolocation of Flying Insects by Bats" [1960]: 141-154) gave the following distances (in centimeters) at which a bat first detected a nearby insect: $$ \begin{array}{lllllllllll} 62 & 23 & 27 & 56 & 52 & 34 & 42 & 40 & 68 & 45 & 83 \end{array} $$ a. Calculate and interpret the mean distance at which the bat first detects an insect. b. Calculate the sample variance and standard deviation for this data set. Interpret these values.

The report titled "State of the News Media \(2013^{\text {" }}\) (Pew Research Center, May 7,2013 ) included the weekday circulation numbers for the top 20 newspapers in the country. Here are the data for the 6 months ending September 2012: $$ \begin{array}{rrrrr} 2,293798 & 1,713,833 & 1,613,866 & 641,369 & 535,875 \\ 529,999 & 522,868 & 462,228 & 432,455 & 412,669 \\ 411,960 & 410,130 & 392,989 & 325,814 & 313,003 \\ 311,504 & 300,277 & 296,427 & 293,139 & 285,088 \end{array} $$ a. Calculate and interpret the value of the median of this data set. b. Explain why the median is preferable to the mean for describing center for this data set. c. Explain why it would be unreasonable to generalize from this sample of 20 newspapers to the population of all daily newspapers in the United States.

The Insurance Institute for Highway Safety (www.iihs .org, June 11,2009 ) published data on repair costs for cars involved in different types of accidents. In one study, seven different 2009 models of mini- and micro-cars were driven at 6 mph straight into a fixed barrier. The following table gives the cost of repairing damage to the bumper for each of the seven models. $$ \begin{array}{|lc|} \hline \text { Model } & \text { Repair Cost } \\ \hline \text { Smart Fortwo } & \$ 1,480 \\ \text { Chevrolet Aveo } & \$ 1,071 \\ \text { Mini Cooper } & \$ 2,291 \\ \text { Toyota Yaris } & \$ 1,688 \\ \text { Honda Fit } & \$ 1,124 \\ \text { Hyundai Accent } & \$ 3,476 \\ \text { Kia Rio } & \$ 3,701 \\ \hline \end{array} $$ a. Calculate and interpret the value of the median for this data set. b. Explain why the median is preferable to the mean for describing center in this situation.

Data on weekday exercise time for 20 males, consistent with summary quantities given in the paper "An Ecological Momentary Assessment of the Physical Activity and Sedentary Behaviour Patterns of University Students" (Health Education Journal [2010]\(: 116-125),\) are shown below. Calculate and interpret the values of the median and interquartile range. $$ \begin{array}{rrrrrrrrr} 43.5 & 91.5 & 7.5 & 0.0 & 0.0 & 28.5 & 199.5 & 57.0 & 142.5 \\ 8.0 & 9.0 & 36.0 & 0.0 & 78.0 & 34.5 & 0.0 & 57.0 & 151.5 \\ 8.0 & 0.0 & & & & & & & \end{array} $$

The article "The Wedding Industry's Pricey Little Secret \(^{n}\) (June \(12,2013,\) www.slate.com, retrieved April \(19,\) 2017 ) stated that the widely reported average wedding cost is grossly misleading. The article reports that in \(2012,\) the average wedding cost was $$\$ 27,427$$ and the median cost was $$\$ 18,086$$ a. What does the large difference between the mean cost and the median cost tell you about the distribution of wedding costs in \(2012 ?\) b. Do you agree that the average wedding cost is misleading? Explain why or why not. c. The article also states "the proportion of couples who spent the 'average' or more was actually a minority." Do you agree with this statement? Explain why or why not using the reported values of the mean and median wedding cost.