3. exercises_descriptive

Introductory Statistics Explained (1.11 Draft) Exercises Descriptive Statistics © 2022, 2023, 2024 Jeremy Balka J.B.’s strongly suggested exercises: 1 , 4 , 7 , 8 , 9 , 10 , 11 , 16 , 17 , 20 , 21 , 22 , 24 , 25 , 36 , 46 , 49 NB The titles and numbers of sections may not (yet) sync up with what is in the text. 1 Plots for Qualitative and Quantitative Variables 1.1 Plots for Qualitative Variables 1. A Finnish study 1 investigated a possible association between the gender of convicted murderers and their relationship to the victim. A random selection of 91 female murderers and a random selection of 91 male murderers were obtained, and the results are illustrated in Figure 1 . Gender Male Female Total Acquaintance 61 37 98 Partner 22 32 54 Family Member 4 18 22 Stranger 4 4 8 Total 91 91 182 Male Offenders Female Offenders 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Relative Frequency Acquaintance Partner Family Stranger Figure 1: Gender of murderer and their relationship to the victim in Finnish murders. (a) For male murderers, summarize the distribution of the relationship between the murderer and the victim. (b) Describe the observed di ff erences in the distributions of the relationship to the victim between male and female murderers. (c) Can we be certain that the observed di ff erences in the samples of male and female murderers reflect the true di ff erences in the population distributions? (d) Sketch a pie chart to illustrate the distribution of the relationship to the victim for male mur- derers. (Sketch a rough plot – it does not need to be very accurate.) 1 Häkkänen et al. (2009). Gender di ff erences in Finnish homicide o ff ence characteristics. Forensic Science International , 186:75–80. 1 connections more intent to murder females more likely to murderclose relationships yes b c random d Aquintone ner igr

1.2 Plots for Quantitative Variables 2 Numerical Measures 2.1 Summation Notation 2. Suppose we have the following sample data set: 8, 14, 22, - 5 (a) What is the value of x 3 ? (b) What is 3 X i =1 x i ? (c) What is P x i ? (d) What is P x 2 i ? 2.2 Measures of Central Tendency 2.2.1 Mean, Median, and Mode 3. Suppose we have a sample of 5 observations: 1, 5, 2, - 3 , 987. (a) What is the mean? (b) What is the median? (c) What is the mode? (d) If the extreme value is removed, what is the mean? (e) If the extreme value is removed, what is the median? 4. A random sample of 4 Canadian male newborns revealed the following birth weights, in grams: 2870 , 2620 , 3120 , 3620 (a) What is the mean birth weight? (b) What is the median birth weight? (c) What are the units of the mean? (d) What are the units of the median? (e) In this situation, which is the more appropriate measure of central tendency, the mean or the median? 2.2.2 Other Measures of Central Tendency 5. What would be an advantage of using the trimmed mean instead of the untrimmed mean? What would be a disadvantage? 2.3 Measures of Variability 6. Consider the following sample of 4 observations: 18, 8, 3, 17. (a) What are the deviations? (b) What is the sum of the deviations? x x x x x 22 0 Xi x t atx 8 14 22 44 3 s 987 1928.4 no a onur more sonone 92 2 1.5 grams Either b c no outliers in TmT T.mn x x x x x ̅ 18,413 17 9 x ̅ der a s is always 0

temp2 Frequency 0 10 20 30 40 0 20 40 60 Score on Test Frequency . Figure 2: Scores on a hard test ( ¯ x = 22 . 4 , s = 7 . 2 ). (b) What would Chebyshev’s theorem tell us about the proportion of observations that are within 7.2 of 22.4? (c) What would Chebyshev’s theorem tell us about the proportion of observations that are within 14.4 of 22.4? (d) What would Chebyshev’s theorem tell us about the proportion of observations that are within 21.6 of 22.4? 13. Consider the histogram given in Figure 3 . (a) Would the empirical rule apply to this data? (b) Would Chebyshev’s theorem apply to this data? Figure 3: A distribution that is skewed to the right. 2.3.2 Why divide by n - 1 in the sample variance formula? 14. Why do we divide by n - 1 in the sample variance formula? 15. Suppose we have a sample of size n = 87 , and the population mean and variance are unknown. How many degrees of freedom are there for estimating the variance? K Sd 4 1 1 f 0 only useful if so 42 1 0.7s at Least 75 of obs fall within a sd of the mean k3 1 f 0.889 at least 89 of obs Fall within 3sd No not mound shape proportion o

2.4 Measures of Relative Standing 2.4.1 Z -scores 16. A random sample of 4 Canadian male newborns revealed the following birth weights, in grams: 2870 , 2620 , 3120 , 3620 In Exercise 7 , we found that for these four observations: ¯ x = 3057 . 5 and s = 426 . 9563 . (a) What are the 4 z -scores? (b) What is the mean of the 4 z -scores? (c) What is the standard deviation of the 4 z -scores? (d) If a newborn male baby had a z -score of 4.6, what does that tell us about the baby’s weight? (e) If a newborn male baby had a z -score of - 0 . 4 , what does that tell us about the baby’s weight? 17. Todd has always had a dream of becoming a medical doctor. After doing well in an introductory statistics course, Todd decides to write the MCAT. His score on the test corresponded to a z -score of 3.0. Suppose that scores on the test have a distribution that is mound-shaped (approximately normal). Which of the following statements are true? (a) The z -score is a unitless quantity. (b) Todd’s score was 3 standard deviations greater than the mean score. (c) Todd scored worse than approximately 1/3 of the test writers. (d) Todd’s score was better than average. 2.4.2 Percentiles 18. A sample of 8 boxes of Kellogg’s All Bran was collected at a grocery store. The boxes had a nominal weight of 675 grams. The weight (in grams) of the cereal in each box was recorded, with the following results: 684 , 684 , 686 , 691 , 691 , 686 , 691 , 684 (The weights were recorded after discarding the box and the bag, so they represent the weight of only the cereal. Real data, collected by JB.) (a) Use the method outlined in the text to calculate the 80th percentile of the weights. (b) Use the method outlined in the text to calculate the 25th percentile of the weights. 19. The 90th percentile of heights of adult females in the United States is closest to which one of the following? (a) 90 cm. (b) 122 cm. (c) 143 cm. (d) 171 cm. (e) 200 cm. 2 is x ̅ 0.439 1.025 0.146 1.317 ALWAYS ALWAYS 1 T timost thebest warfarin sina.gg ordered 684 684,684,686686,691,691,69 b I If est siingeish To percentile

4 Linear Transformations 22. A certain sample has a mean of 100, a median of 90, and a standard deviation of 20. Suppose that each observation has 5 added to it, and the result is multiplied by 10. (a) What is the mean of the new values? (b) What is the variance? (c) What is the standard deviation? (d) What is the median? 23. Suppose we have a sample of 7 values, and we calculate the usual summary statistics. If we add 5,000 to each of these 7 sample values, then recalculate the summary statistics, which of the following quantities would change? mean, median, Q 3 , IQR, variance, standard deviation 5 Chapter Exercises 5.1 Basic Calculations 24. Consider the following sample of 4 numbers: 52, - 2 , 8 , 107 . (a) What is the mean? (b) What is the median? (c) What is the variance? (d) What is the standard deviation? (e) What is the range? 25. Consider the following sample of 7 observations: 3.1, 8.2, 9.6, - 1 . 7 , 8.4, 8.8, 21.1 (a) What is the mean? (b) What is the median? (c) What is the third quartile ( Q 3 )? (d) What is the interquartile range? (e) What is the variance? (f) What is the standard deviation? (g) Are any of these observations outliers according to the 1.5*IQR rule? 26. Consider the following sample data set: 32 , 36 , 1 , 4 , 89 , 18 . (a) What is the mean? (b) What is the median? (c) What is the value of Q 1 ? (d) What is the value of the interquartile range? (e) What is the variance? (f) What is the standard deviation? 27. Consider the following sample data set: - 4 , 6 , - 12 , 14 , 742 x 5 10 100 5 0 11050 202 1 5,5 8 90 5 0 950 addingfoesfn.se fFjt measures only meseurs of rotationposition 41.25 38471 107 21 48 1.3 3.1 8.28.48.8 9.621 8 I s 3 48.79 6.98 i msn.si 6 78 15 2 ypa.gg L's 5 36 1036.4 32

(a) What is the mean of the full data set? (b) What is the median of the full data set? (c) What is the standard deviation of the full data set? (d) What is the variance of the full data set? (e) If the outlier is discarded, what is the mean? (f) If the outlier is discarded, what is the median? (g) If the outlier is discarded, what is the standard deviation? 28. Consider the following computer output, illustrating a stemplot. Decimal point is at the colon -1 : 7110 -0 : 76632 0 : 03335 1 : 01389 2 : 4 (a) What is the mean? (b) What is the median? (c) What is the variance? (d) What is the standard deviation? (e) What is the range? 29. Consider the following split-stem stemplot: (The decimal is at the colon. You should be able to recognize that the largest observation is 5.8.) 0 : 0000000000000000000111111222222233334444444 0 : 557777888 1 : 001111224444 1 : 5555778999 2 : 011223 2 : 55566689 3 : 0 3 : 5678 4 : 4 : 7789 5 : 5 : 668 (a) Is the mean greater than the median? (b) Does the distribution show any skewness? If so, in what direction is it skewed? (c) What is the value of the di ff erence between the fifth and first values in the five-number summary? 30. Consider the following stemplot:

-10 0 10 20 30 40 A -10 0 10 20 30 40 B 0 5 10 15 20 C Figure 7: Which boxplot corresponds to the histogram? (a) What must the value of the fifth deviation be? (b) What was the sample mean? (c) What is the standard deviation? 33. If the smallest 4 values in a sample with n = 5 are equal, and the range is equal to 10, what is the value of the variance? 34. Consider the frequency histogram shown in Figure 8 . Which of the following statements are true? 0 1 2 3 4 5 0 10 20 30 40 Frequency x Figure 8: A frequency histogram. (a) The median is less than 4. (b) The mean is less than the median. (c) The IQR is greater than 5. (d) All the numbers in the five number summary are positive. (e) This distribution is approximately symmetric. (f) The value of Q 1 - Median would have the same absolute value as Q 3 - Median. (g) There are no obvious outliers. 35. Consider the boxplots in Figure 9 . large SD shows 2 variables i right skew sum must 0 2.313.71.2 0.7 1.9 stdev.is 1.9 imposs IE j iiiiiiii assa sr or an 2,212,2112 var 20 s dontmatter see P Timedian methmedian Hahn T F lar 0 1 t.mn If Eat s 3 answers 8 be rather

0 1 2 3 4 5 A 1 2 3 4 5 B 2 3 4 5 6 7 C Figure 9: Which boxplot represents the histogram data? (a) Which one of the boxplots corresponds to the histogram in Figure 8 ? (b) Which boxplot has the largest median? (c) Which boxplot has the largest interquartile range? (d) Which boxplot contains the largest number of outliers? 36. Suppose we have a sample of 12 observations, and we calculate the summary statistics for these values. (a) If we add 100,000 to the largest observation, which of the following statistics would change? Mean, median, variance, standard deviation, interquartile range (b) If we add 100,000 to the largest observation, and subtract 100,000 from the smallest observation, which of the following statistics would change? Mean, median, variance, standard deviation, interquartile range 37. Suppose there is a data set of 100 observations that has a standard deviation equal to 0. Which of the following statements are true? (a) All the numbers in the 5 number summary must be equal. (b) There are no negative deviations. (c) The variance and standard deviation are equal. (d) The mean and median are equal. (e) If a single observation from this data set had a positive constant added to it, then the IQR, minimum, and median would not change. 38. Suppose we have a sample data set with 4 observations, where ¯ x = 50 and s = 0 . Suppose we add one more observation to this data set, and we recalculate the mean and standard deviation. If the added observation equals 50.0, what will the standard deviation of the 5 number data set be? 39. Which of the following statements are true? (a) The standard deviation is always greater than the median. (b) The variance is always greater than the standard deviation. (c) If a distribution is skewed to the left, then the mean will always be less than the standard deviation. (d) If all values in a sample are within 2.0 of each other, then the standard deviation will be less than 2.0. C A M 1 right skew B A c a a 1000 median 14,6 7 mid Iar resistant to outliers Q Qs do not chs esinifntthtkiik.ws increases morevariety resistant to utliers 50 0 555,5 etc sp o if all data r equal T T T T T rail vines so so so so I sa nor so sa larger than var F not comparable T SD av distance from man so w ̅ men 1 if sample Itn e w ̅ sample 10,2 Sd 1 41

20 40 60 80 100 0 40 80 120 Final Grade Frequency Figure 10: Final grades in an introductory statistics course. (b) What is the approximate value of Q 1 ? (c) What is the approximate value of the interquartile range? (d) Would the mean be greater or less than the median? (e) What is the approximate value of the standard deviation? 46. Figure 11 is a histogram of grades on a major assignment in a large introductory statistics course. (This is real data from one of JB’s courses.) Of the 206 students in the course, twelve students did not hand in the assignment and received a grade of 0. The average grade for all students was 18.8 out of 25. 0 5 10 15 20 25 0 5 10 15 20 25 Grade on Assignment Frequency Figure 11: Grades on a major assignment. (a) Describe the distribution. (b) Reporting the mean grade for all students might be slightly misleading in this setting. Suggest a better numerical summary of this data. 47. The insect Megamelus scutellaris can feed on the sap of the water hyacinth, and it has been used as a method of biocontrol for this invasive plant species. M. scutellaris mates on the water hyacinth, and females create oviposition scars on the plant, laying one or more eggs in each scar. Sosa et al. ( 2005 ) investigated several aspects of the reproductive cycle of this insect. In one aspect of this study, the number of eggs laid per scar was measured. The observed number of eggs per scar for 359 scars is given in Table 1 .

1 egg/scar 2 eggs/scar 3 eggs/scar 4 eggs/scar 142 194 9 14 Table 1: Observed number of M. scutellaris eggs per scar. (a) What is the mean number of eggs per scar? (b) What is the median number of eggs per scar? (c) What is the standard deviation of the number of eggs per scar? 48. As part of an investigation by Fink et al. ( 2007 ), 32 male student volunteers at a university in Germany had their hand grip strength measured (in kilograms force (kgf)). The results are illustrated in Figure 12 . 30 35 40 45 50 55 60 65 0 2 4 6 8 Hand Grip Strength (kgf) Frequency 35 40 45 50 55 60 Hand Grip Strength (kgf) Figure 12: Plots of the grip strengths of 32 male student volunteers. (a) Describe the distribution of the grip strengths. (b) Estimate the 90th percentile of the grip strengths in this data set. (c) Estimate the 90th percentile of the grip strengths of adult Germans. (d) For this data, is the mean or the median the more appropriate measure of central tendency? (e) Give rough estimates of the mean and median. (f) Give a rough estimate of the standard deviation. 49. Tantius et al. ( 2014 ) investigated tensile properties of human umbilical cords. In one part of the study, 23 human umbilical cords were stretched until they broke, and their elongation (% increase in length) at the breaking point was recorded. (The breaking point of umbilical cords is of interest in forensic science, as mothers accused of killing a newborn might claim that an umbilical cord broke during childbirth and the newborn baby bled to death.) The elongation percentages are illustrated in Figure 13 . (a) Describe the distribution of the elongation percentages. (b) The mean elongation percentage of these 23 umbilical cords is 36.6. Estimate the mean if the outlier were to be removed from the calculations. (c) Should the outlier be omitted from the analysis? (d) The study actually involved 25 umbilical cords, but in two cases the machine did not recognize the break and thus did not record the elongation at the breaking point. These data points were omitted in this plot and in the calculation of the mean. How might this distort the results? 50. Harlioglu et al. ( 2012 ) investigated several characteristics of crayfish in a freshwater lake in Turkey. In this study, twenty-five adult male Astacus leptodactylus crayfish were sampled, and several variables

52. A person is weighing rocks in a lab in the United States. They are new to the United States from Canada, and they have been taking all of the measurements in kilograms. The sample mean and standard deviation of the weight of the rocks (in kilograms) were found to be 15.10 and 1.70, respectively. At this point the person realizes two important points: 1) The scale wasn’t calibrated properly, and every measurement in the data set should have 2.00 kilograms added to it. 2) Their boss wants the mean and the standard deviation in pounds, not kilograms (1 kg = 2.20 pounds). The person correctly realizes that they must add 2.00 to all of their measurements, then multiply the result by 2.20. What are the mean and standard deviation of the correct weights in pounds? References Fink et al. (2007). Male facial appearance signals physical strength to women. American Journal of Human Biology , 19:82–87. Häkkänen et al. (2009). Gender di ff erences in Finnish homicide o ff ence characteristics. Forensic Science International , 186:75–80. Harlioglu et al. (2012). An investigation on the sperm number and reproductive parameters of males in wild caught freshwater crayfish ( Astacus leptodactylus , eschscholtz). Animal Biology , 62:409–418. Qu et al. (2011). Sexual dimorphism and female reproduction in two sympatric toad-headed lizards, Phrynocephalus frontalis and P. versicolor (Agamidae). Animal Biology , 61:139–151. Sosa et al. (2005). Life history of Megamelus scutellaris with description of immature stages (Hemiptera: Delphacidae). Annals of the Entomological Society of America , 98(1):66–72. Tantius et al. (2014). Experimental studies on the tensile properties of human umbilical cords. Forensic Science International , 236:16–21.