Exercise 2.1: Comparing Test Scores Adapted from Proceedings of CERME 6, January 28th-February 1st 2009, Lyon France © INRP 2010 <www.inrp.fr/editions/cerme6>
Math Class 10a and Math Class 10b were given the same test. In class 10a, there are 30 students and in class 10b there are 29 students. The two box plots show the distributions of the scores of the students in these two classes on this test:
What differences do you see in the test scores for these two classes? What recommendations might you make to the teacher, based on these differences? Support your response with evidence from the box plots and your understanding of box plots.
Exercise 2.2: Make Your Own Exam Scores Adapted from Allan J. Rossman, Beth L. Chance, and Robin H. Lock, Workshop Statistics: Discovery with Data and Fathom (Key College Publishing, 2001), 87, 116. For each of the following properties, construct you own data set of 10 hypothetical exam scores (integers between 0 and 100, inclusive) satisfying that property. Describe how you arrived at each answer and include a dotplot of the distribution. You should use Fathom to help you with this task.
(a) 90% of the scores are greater than the mean. (b) The IQR is zero, and the median is less than the mean. (c) Less than half of the test scores are within 1 SD of the mean. (d) The SD is half as large as the range.
Exercise 2.3: ATM Withdrawals Adapted from Allan J. Rossman, Beth L. Chance, and Robin H. Lock, Workshop Statistics: Discovery with Data and Fathom (Key College Publishing, 2001), 116-117 LittleTown Bank is monitoring the withdrawals customers make from the ATMs it has in three locations around the town. They would like to know how withdrawals vary among the three ATMS. They collect 50 random samples of withdrawals from each ATM on a given day. The mean, median, and standard deviation for the withdrawals are summarized in the table below. The data for this problem can be found in the Excel file HypoATM.
HypoATM m achine 1 m achine 2 m achine 3
70 30.3046
70
70 30.3046
70
70 30.3046
70 S1 = mean S2 = stdDev S3 = median
Notice that the measures of center and standard deviation are identical for these distributions. Does this mean the distributions are identical? Justify your response with evidence in the data and your understanding of mean, median, and standard deviation.
Exercise 2.4: Larger Mean or Larger Median? A test to measure aggressive tendencies was given to a group of teenage boys who were members of a street gang. The test is scored from 10 to 60, with a high score indicating more aggression. The histogram below represents the results for these 28 boys. Which do you think will be larger, the mean or the median score? Or do you think they will be similar? Explain without directly calculating the mean or the median.
Exercise 2.5: Higher Standard Deviation For each pair of graphs, determine which graph (if either) has the higher standard deviation. Justify your choices without calculating the exact standard deviations.
