Download Statistical Reasoning Exercises: Analyzing Data and Probability and more Exams Probability and Statistics in PDF only on Docsity!
Name Course Tutor Date A) Analyzing Students’ Statistical Reasoning [A1] Data Displays Your 30 students measured the height of a statue. Two students created displays of the data they collected. The two displays show the same measurements. Display A: Display B: Identify two differences between the displays that affect what you can see in the data and explain what each difference (a) helps you see in the data or (b) makes it harder to see in the data. Be specific: point to exactly where in the display you can see that difference. Difference # In display A, we can’t clearly tell how many students got each value, but this can readily be found in display B. For example, we can see that 7 students had their height measure as 64-65 in A, but we don’t know how many recorded 64 and how many 65, or even whether there was a 65 or 64 reading. However, in B, it is clear 3 students recorded 64 and 4 students recorded 65. The Height of the Statue in Meters The Height of the Statue in Meters
Difference # Display B is better at showing where most of the measurements are than display A. Here, display B gives us the exact frequency for each height, enabling us to know the precise measurement for each height. In A, one can’t clearly tell how many measurements they got for each height since they have been grouped in 2’s. For example, in A one can’t know whether the highest point is 64, 65,66 or 67, but in B it is clear it is 65 and 66. [A2] Interpretations of the Mean You have been teaching your 6th-grade students two different interpretations of the mean : (1) mean as a balance point, and (2) mean as a ‘fair share’ (Van de Walle calls this “leveling”). You’re interested in which interpretation they use as they solve problems. You’ve given your students the following problem: There are five 9th-grade math classes at a local middle school. I wrote down the number of students in each of those 5 classes. Three of the values are below, but I forgot to write down the other two. The mean of all five values is 26. What could the other two values be? (a) Isaac correctly solved the problem in the following way: “The mean is 26 and there are 5 values, so I did 5 x 26, which is 130. That means if you added up all 5 values, you’d get 130. So far we have 24, 27 and 29. If you add those up, you get 80. 130 – 80 is 50, so the other two values have to add up to 50. You can choose any numbers that add up to 50. I just chose 24 and 26.” Does Isaac’s strategy reflect a “fair share” or a “balance point” interpretation of the mean? Explain, referencing what Isaac did. In your explanation, you also should state what the mean tells us about this data in terms of students and classrooms. Isaac’s strategy reflects a ‘balancing point’ interpretation of the mean. Upon adding the three given values and subtracting the sum total from the total SUM, he remains with 50. There are many numbers that can be added to generate the sum, but he chose to assume them and even not choose two 25s of equal share, but chooses 24 and 26 which are not equal, and this thus can be attributed to trying to just balance the equation.
[A3] Measures of Variation Two classes of students measured the height of a young tree. Here are their results: Ms. Shahan’s Class (Median 58; Mean 58.1053) Dr. Jackson’s Class (Median 58; Mean 58.47) The students were interested in figuring out which class’ measurements were more precise. Here are Leah and Mark’s methods for measuring how precise each class’ measurements were. Leah “I subtracted the lowest measurement from the highest measurement. That was 30 for Ms. Shahan’s class and 18 for Dr. Jackson’s class, so Dr. Jackson’s class was more precise.” Mark “I found the median of each set of measurements. The median was 58 for both classes. Then I figured out how many measurements were close to 58. In Ms. Shahan’s class, there were 12 measurements between 56 and 60. In Dr. Jackson’s class there were 8 measurements between 56 and 60. There were more measurements close to the median in Ms. Shahan’s class, so her class was more precise.” (1) What is a strength of Mark’s method for measuring the precision of the measurements? Explain why this is a strength. Mark chose to use a median mark and this is the point of strength. Here, the values make sense as they don’t vary much from each other, and this weakens the influence of an outlier mark.
(2) What are one limitation to Mark’s method for measuring the precision? Explain why this is a limitation. It will not get him the exact value. Median mostly offers a middle value, and in the event one value is not right, it will give him wrong value. Using the mean is the best approach. Also, for the observation are from two groups, the median of the pooled group cannot be expressed in terms of the individual medians of those groups. (3) Which class do you think measurement more precisely? Explain your reasoning. You should justify your choice with at least one measure of variation and you should explain how you calculated that measure of variation. I think Mrs. Shahan’s class did the measurements more precisely. Most of its measurements are realized to concentrate between height 55 and height 60 inches, hence depicting some consistency. Let us consider Interquartile range as a measure of variation. This shows how data is spread around the mean value. Inter-quartile range (IQR) = Q 3 – Q 1 Our set of data arranged chronologically is: 50, 52, 53, 55 , 55, 56, 56, 57, 57, 58 , 58, 59. 59, 59, 59 , 60, 60, 61, 80. The median is 58, and therefore the first quartile Q 1 is the median of first half and the second quartile Q 2 the median of the second half. Q 1 = 55 and Q 2 = 59 Therefore, IQR =59-55 = 4. B) Connecting Statistics & Probability – Big Ideas [6 points] You showed your 7th-grade students a “mystery spinner” and asked them to set up an experiment to figure out if it was actually Spinner A, Spinner B, or Spinner C below. Two of your students ran experiments: Tesha: I spun the mystery spinner 120 times and recorded what percentage was red. Then I did that 49 more times. In total, I collected 50 samples, each of 120 spins. Chandra: I spun the mystery spinner 1000 times and recorded what percentage was red. Then I did that 49 more times. In total, I collected 50 samples, each of 1000 spins.
is this student demonstrating? Explain it in your own words or using language from the class or from readings. She has let the prior results of the spin influence her judgement and forgotten that spinning is something random or of chances. The believes all events are likely equal.
