Ismor Fischer, 5/26/

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1 Ismor Fischer, 5/6/ Problems. Follow the instructions in the posted R code folder ( for this problem, to reproduce the results that appear in the lecture notes for the Memorial Union age data.. A numismatist (coin collector) has a large collection of pennies minted between the ears , when the were made of bronze: 95% copper, and 5% tin and zinc. (Toda, pennies have a 97.5% zinc core; the remaining.5% is a ver thin laer of copper plating.) The ear the coin was minted appears on the obverse side (i.e., heads ), sometimes with a letter below it, indicating the cit where it was minted: D (Denver), S (San Francisco), or none (Philadelphia). Before 959, a pair of wheat stalks was depicted on the reverse side (i.e., tails ); starting from that ear, this image was changed to the Lincoln Memorial. The overall condition of the coin follows a standard grading scale Poor (PO or PR), Fair (FA or FR), About Good (AG), Good (G), Ver Good (VG), Fine (F), Ver Fine (VF), Etremel Fine (EF or XF), Almost Uncirculated (AU), and Uncirculated or Mint State (MS) which determines the coin s value. (a) Using this information, classif each of the following variables as either numerical (specif continuous or discrete) or categorical (specif nominal: binar, nominal: not binar, or ordinal). Amount of zinc Image on reverse Year minted Cit minted Condition (b) Suppose the collector accidentall drops 000 pennies. Repeat the instructions in (a) for the variables Number of heads face-up Proportion of heads face-up 3. Sketch a dotplot (b hand) of the distribution of values for each of the data sets below, and calculate the mean, variance, and standard deviation of each. U:,, 3, 4, 5, 6, 7 X:, 3, 4, 4, 4, 5, 6 Y: 3, 4, 4, 4, 4, 4, 5 Z: 4, 4, 4, 4, 4, 4, 4 What happens to the mean, variance, and standard deviation, as we progress from one data set to the net? What general observations can ou make about the relationship between the standard deviation, and the overall shape of the corresponding distribution? In simple terms, wh should this be so? 4. Useful Properties of Mean, Variance, Standard Deviation (a) Suppose that a constant b is added to ever value of a data set {,, 3,, n }, to produce a new data set { b, b, 3 b,, n b }. Eactl how are the mean, variance, and standard deviation affected, and wh? (Hint: Think of the dotplot.) (b) Suppose that ever value in a data set {,, 3,, n } is multiplied b a nonzero constant a to produce a new data set { a, a, a3,, a n }. Eactl how are the mean, variance, and standard deviation affected, and wh? Don t forget that a (and for that matter, b above) can be negative! (Hint: Think of the dotplot.)

2 Ismor Fischer, 5/6/ During a certain winter in Madison, the variable X = Temperature at noon ( F) is measured ever da over two consecutive weeks, as shown below. Sun Mon Tues Wed Thurs Fri Sat Week Week (a) Calculate the sample mean temperature and sample variance s for Week. (b) Without performing an further calculations, determine the mean temperature and sample variance s for Week. [Hint: Compare the Week temperatures with those of Week, and use the result found in 4(b).] Confirm b eplicitl calculating. 6. A little practice using R: First, tpe the command pop = :00 to generate a simulated population of integers from to 00, and view them (read the intro to R to see how). (a) Net, tpe the command.vals = sample(pop, 5, replace = T) to generate a random sample of n = 5 values from this population, and view them. Calculate, without R, their sample mean, variance Show all work! s, and standard deviation s. (b) Use R to calculate the sample mean in two was: first, via the sum command, then via the mean command. Do the two answers agree with each other? Do the agree with (a)? If so, label this value bar. Include a cop of the R output in our work. (c) Use R to calculate the sample variance in two was: first, via the sum command, then via the var command. Do the two answers agree with each other? Do the agree with (a)? If so, label this value s.sqrd. Include a cop of the R output in our work. (d) Use R to calculate the sample standard deviation in two was: first, via the sqrt command, then via the sd command. Do the two answers agree with each other? Do the agree with (a)? Include a cop of the R output in our work. 7. (You ma want to refer to the Rcode folder for this problem.) First pick n = 5 numbers at random,, 3, 4, 5, and calculate their sample mean and standard deviation s. (a) Compute the deviations from the mean i for i,,3, 4,5, and confirm that their sum = 0. The idea behind this problem will be important in Chapter 4. (b) Now divide each of these individual deviations b the standard deviation s. These new values i z, z, z3, z4, z 5 are called standardized values, i.e., zi s, for i,,3, 4,5. Calculate their mean z and standard deviation s z. Repeat several times. What do ou notice? (c) Wh are these results not surprising? (Hint: See problem 4.)

3 Ismor Fischer, 5/6/ (a) The average score of a class of n 0 students on an eam is 90.0, while the average score of another class of n 30 students on the same eam is If the two classes are pooled together, what is their combined average score on the eam? (b) Suppose two other classes one with n 4 students, the other with n 44 students have the same mean score, but with standard deviations s 7.0 and s 0.0, respectivel. If these two classes are pooled together, what is their combined standard deviation on the eam? (Hint: Think about how sample standard deviation is calculated.) 9. (Hint: See page.3-) A random sample of n = 90 people is grouped according to age in the frequenc table below: Age Group Frequenc [0, 0) 9 [0, 5) 7 [5, 65] 54 (a) Calculate the group mean age and group standard deviation. Epress in ears and months. (b) Construct a relative frequenc histogram. (c) Construct a densit histogram. (d) What percentage of the sample falls between 5 and 35 ears old? (e) Calculate the group quartile ages Q, Q, Q 3. Epress in terms of ears and months. (f) Calculate the range and the interquartile range. Epress in terms of ears and months. 0. For an 0,,,, n, consider a data set,, 3,, n consisting entirel of n ones and ( n ) zeros, in an order. For eample, {,,,, 0, 0,, 0 }. Also denote the sample proportion of ones b p n. (a) How man such possible data sets can there be? (b) Construct a relative frequenc table for such a data set. (c) Show that the sample mean p. (d) Show that the sample variance s p ( p ). n n

4 Ismor Fischer, 5/6/ (a) Consider the sample data {0,0,0,,0, 60, 60, 60,, 60}, where half the values are 0 and half the values are 60. Complete the following relative frequenc table for this sample, and calculate the sample mean and sample median. p ( ) i 0 60 i (b) Suppose the original dataset is unknown, and onl given in grouped form, with each of the two class intervals shown below containing half the values. Class Interval [0, 0) Relative Frequenc [0, 00) Complete this relative frequenc table, and calculate the group sample mean and group sample median. How do these compare with the values found group in (a)? Sketch the relative frequenc histogram. Sketch the densit histogram. Label the group sample mean and median in each of the two histograms. In which histogram does the mean more accuratel represent the balance point of the data, and wh?. B the end of the semester, Merriman forgets the scores he received on the four quizzes (each worth 00 points) he took in a certain course. He onl remembers that their average score was 80 points, standard deviation 0 points, and that 3 out of the 4 scores were the same. From this information, compute all four missing quiz scores. [Hint: Recall that the i th deviation of a value i from the mean is defined as di i, so that i d i for i,,3, 4. Then use the given information.] Note: There are two possible solutions to this problem. Find them both.

5 Ismor Fischer, 5/6/ Linear Interpolation (A generalization of the method used on page.3-6.) If software is unavailable for computations, this is an old technique to estimate values which are in-between tabulated entries. It is based on the idea that over a small interval, a continuous function can be approimated b a linear one, i.e., constant slope. Column A Column B a b v w v w a b Given two successive entries a and a in the first column of a table, with corresponding successive entries b and b, respectivel, in the second column. For a given value between a and a, we wish to approimate the corresponding value between b and b, or vice versa. Then assuming equal proportions, we have b b b a a a. Show that this relation implies that can be written as a weighted average of b and b. In particular, v b v v b v where the weights are given b the differences v a and v a. Similarl,, w a w w a w, where the weights are given b the differences w b and w b.