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One standard for admission to Redfield College is that the student must rank in the upper quartile of his or her graduating high school class. What is the minimal percentile rank of a successful applicant?

When a distribution is mound-shaped symmetrical, what is the general relationship among the values of the mean, median, and mode?

At Center Hospital there is some concern about the high turnover of nurses. A survey was done to determine how long (in months) nurses had been in their current positions. The responses (in months) of 20 nurses were \(\begin{array}{rrrrrrrrrr}23 & 2 & 5 & 14 & 25 & 36 & 27 & 42 & 12 & 8 \\ 7 & 23 & 29 & 26 & 28 & 11 & 20 & 31 & 8 & 36\end{array}\) Make a box-and-whisker plot of the data. Find the interquartile range.

Critical Thinking: Data Transformation In this problem, we explore the effect on the standard deviation of adding the same constant to each data value in a data set. Consider the data set \(5,9,10,11,15\). (a) Use the defining formula, the computation formula, or a calculator to compute \(s\). (b) Add 5 to each data value to get the new data set \(10,14,15,16,20 .\) Compute \(s\). (c) Compare the results of parts (a) and (b). In general, how do you think the standard deviation of a data set changes if the same constant is added to each data value?

In this problem, we explore the effect on the mean, median, and mode of multiplying each data value by the same number. Consider the data set \(2,2,3,6,10\). (a) Compute the mode, median, and mean. (b) Multiply each data value by \(5 .\) Compute the mode, median, and mean. (c) Compare the results of parts (a) and (b). In general, how do you think the mode, median, and mean are affected when each data value in a set is multiplied by the same constant? (d) Suppose you have information about average heights of a random sample of airplane passengers. The mode is 70 inches, the median is 68 inches, and the mean is 71 inches. To convert the data into centimeters, multiply each data value by \(2.54\). What are the values of the mode, median, and mean in centimeters?

In this problem, we explore the effect on the standard deviation of multiplying each data value in a data set by the same constant. Consider the data set \(5,9,10,11,15 .\) (a) Use the defining formula, the computation formula, or a calculator to com- pute \(s .\) (b) Multiply each data value by 5 to obtain the new data set \(25,45,50,55,75\). Compute s. (c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constant \(c\) ? (d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be \(s=3.1\) miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calculations? Given 1 mile \(=1.6\) kilometers, what is the standard deviation in kilometers?

One indicator of an outlier is that an observation is more than \(2.5\) standard deviations from the mean. Consider the data value 80 . (a) If a data set has mean 70 and standard deviation 5 , is 80 a suspect outlier? (b) If a data set has mean 70 and standard deviation 3, is 80 a suspect outlier?

Kevlar epoxy is a material used on the NASA Space Shuttle. Strands of this epoxy were tested at the \(90 \%\) breaking strength. The following data represent time to failure (in hours) for a random sample of 50 epoxy strands (Reference: \(\mathrm{R} . \mathrm{E} .\) Barlow, University of California, Berkeley). Let \(x\) be a random variable representing time to failure (in hours) at \(90 \%\) breaking strength. Note: These data are also available for download online in HM StatSPACETM. \(\begin{array}{llllllllll}0.54 & 1.80 & 1.52 & 2.05 & 1.03 & 1.18 & 0.80 & 1.33 & 1.29 & 1.11 \\ 3.34 & 1.54 & 0.08 & 0.12 & 0.60 & 0.72 & 0.92 & 1.05 & 1.43 & 3.03 \\ 1.81 & 2.17 & 0.63 & 0.56 & 0.03 & 0.09 & 0.18 & 0.34 & 1.51 & 1.45 \\ 1.52 & 0.19 & 1.55 & 0.02 & 0.07 & 0.65 & 0.40 & 0.24 & 1.51 & 1.45 \\ 1.60 & 1.80 & 4.69 & 0.08 & 7.89 & 1.58 & 1.64 & 0.03 & 0.23 & 0.72\end{array}\) (a) Find the range. (b) Use a calculator to verify that \(\Sigma x=62.11\) and \(\Sigma x^{2}=164.23\). (c) Use the results of part (b) to compute the sample mean, variance, and standard deviation for the time to failure. (d) Use the results of part (c) to compute the coefficient of variation. What does this number say about time to failure? Why does a small CV indicate more consistent data, whereas a larger CV indicates less consistent data? Explain.

The Hill of Tara in Ireland is a place of great archaeological importance. This region has been occupied by people for more than 4,000 years. Geomagnetic surveys detect subsurface anomalies in the earth's magnetic field. These surveys have led to many significant archaeological discoveries. After collecting data, the next step is to begin a statistical study. The following data measure magnetic susceptibility (centimeter-gram-second \(\times 10^{-6}\) ) on two of the main grids of the Hill of Tara (Reference: Tara: An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Grid E: \(x\) variable \(\begin{array}{lrrrrrr}13.20 & 5.60 & 19.80 & 15.05 & 21.40 & 17.25 & 27.45 \\ 16.95 & 23.90 & 32.40 & 40.75 & 5.10 & 17.75 & 28.35 \\\ \text { Grid H: } y & \text { variable } & & & & & \end{array}\) \(\begin{array}{lllllll}11.85 & 15.25 & 21.30 & 17.30 & 27.50 & 10.35 & 14.90 \\\ 48.70 & 25.40 & 25.95 & 57.60 & 34.35 & 38.80 & 41.00 \\ 31.25 & & & & & & \\ & & & & & & \end{array}\) (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y\), and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\) (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the magnetic susceptibility on the two grids. Higher numbers indicate higher magnetic susceptibility. However, extreme values, high or low, could mean an anomaly and possible archaeological treasure. (d) Compute the sample coefficient of variation for each grid. Use the CV's to compare the two grids. If \(s\) represents variability in the signal (magnetic susceptibility) and \(\bar{x}\) represents the expected level of the signal, then \(s / \bar{x}\) can be thought of as a measure of the variability per unit of expected signal. Remember, a considerable variability in the signal (above or below average) might indicate buried artifacts. Why, in this case, would a large \(\mathrm{CV}\) be better, 2