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

What symbol is used for the arithmetic mean when it is a sample statistic? What symbol is used when the arithmetic mean is a population parameter?

Clayton and Timothy took different sections of Introduction to Economics. Each section had a different final exam. Timothy scored 83 out of 100 and had a percentile rank in his class of \(72 .\) Clayton scored 85 out of 100 but his percentile rank in his class was \(70 .\) Who performed better with respect to the rest of the students in the class, Clayton or Timothy? Explain your answer.

Consider the following ordered data: $$\begin{array}{lllllll} 2 & 5 & 5 & 6 & 7 & 7 & 8 & 9 & 10 \end{array}$$ (a) Find the \(\operatorname{low}, Q_{1},\) median, \(Q_{3},\) high. (b) Find the interquartile range. (c) Make a box-and-whisker plot.

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?

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?

Consider the numbers 2 3 4 5 5 (a) Compute the mode, median, and mean. (b) If the numbers represent codes for the colors of T-shirts ordered from a catalog, which average(s) would make sense? (c) If the numbers represent one-way mileages for trails to different lakes, which average(s) would make sense? (d) Suppose the numbers represent survey responses from 1 to \(5,\) with \(1=\) disagree strongly, \(2=\) disagree, \(3=\) agree, \(4=\) agree strongly, and \(5=\) agree very strongly. Which averages make sense?

Kevlar epoxy is a material used on the NASA space shuttles. 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: R. 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 at the Companion Sites for this text. $$\begin{array}{lllllllll} 0.54 & 1.80 & 1.52 & 2.05 & 1.03 & 1.18 & 0.80 & 1.33 & 1.29 & 1.11 \\ 3.34 & 1.54 & 0.08 & 0.12 & 0.60 & 0.72 & 0.92 & 1.05 & 1.43 & 3.03 \\ 1.81 & 2.17 & 0.63 & 0.56 & 0.03 & 0.09 & 0.18 & 0.34 & 1.51 & 1.45 \\ 1.52 & 0.19 & 1.55 & 0.02 & 0.07 & 0.65 & 0.40 & 0.24 & 1.51 & 1.45 \\ 1.60 & 1.80 & 4.69 & 0.08 & 7.89 & 1.58 & 1.64 & 0.03 & 0.23 & 0.72 \end{array}$$ (a) Find the range. (b) Use a calculator to verify that \(\Sigma x=62.11\) and \(\Sigma x^{2} \approx 164.23\). (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 \(C V\) indicate more consistent data, whereas a larger \(C V\) 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 4000 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 \(\mathbf{E}: x\) variable $$\begin{array}{ccccccc} 13.20 & 5.60 & 19.80 & 15.05 & 21.40 & 17.25 & 27.45 \\ 16.95 & 23.90 & 32.40 & 40.75 & 5.10 & 17.75 & 28.35 \end{array}$$ Grid H: \(y\) variable $$\begin{array}{lllllll} 11.85 & 15.25 & 21.30 & 17.30 & 27.50 & 10.35 & 14.90 \\ 48.70 & 25.40 & 25.95 & 57.60 & 34.35 & 38.80 & 41.00 \\ 31.25 & & & & & \end{array}$$ (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 \(C V\) 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 \(C V\) be better, or at least more exciting? Explain.

How old are professional football players? The 11th edition of The Pro Football Encyclopedia gave the following information. Random sample of pro football player ages in years: $$\begin{array}{llllllllll}24 & 23 & 25 & 23 & 30 & 29 & 28 & 26 & 33 & 29 \\\24 & 37 & 25 & 23 & 22 & 27 & 28 & 25 & 31 & 29 \\\25 & 22 & 31 & 29 & 22 & 28 & 27 & 26 & 23 & 21 \\\25 & 21 & 25 & 24 & 22 & 26 & 25 & 32 & 26 & 29\end{array}$$ (a) Compute the mean, median, and mode of the ages. (b) Interpretation Compare the averages. Does one seem to represent the age of the pro football players most accurately? Explain.