Answers Final grades Project Part I. Mrs. Pugh recorded all of the semester grades (on a scale of 100) of her 50 Advanced Algebra students in the following dotplot: Mrs. Pugh's Advanced Algebra Grades
60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
Advanced Algebra Grades
A list of the scores:
60,61,63,65,68,68,68,69,69,70,70,72,74,76,76,76,78,79,80,80,80,80,82,82,82,82,84,85,85,86,86,
86,87,88,89,90,90,90,90,90,92,92,92,92,95,95,95,98,99,100 What is the mean and standard deviation of the distribution of individual grades? 1. x = _______________ 2. σ = _______________ Mean is 81.72 (81.7) and the standard deviation is 10.37 (10.4). Identify the following and label the dotplot above 3. µ ± 1σ = _______________ and _______________ 71.3 and 92.1 4. µ
± 2σ = _______________ and _______________ 60.9 and 102.5 5. µ ± 3σ = _______________ and _______________ 50.5 and 112.9 6. Does the data indicate a normal distribution? ______________ yes 7. Why or why not? 66% of the data fell within one standard deviation of the mean, 98% fell within two standard deviations of the mean and 100% fell within 3 standard deviations of the mean. This is very close to the Empirical Rule for a normal distribution 8. We want to know the probability that a student selected randomly from her class would have an “A” (90 or above) in her class. Find the probability using two different methods, including all work. Method 1: 15 dots out of the 50 are 90 or better so fifteen divided by fifty is 0.3 Method 2: Since the Empirical Rule applies, find a z-­‐score of 0.7981. Use the chart you get a probability of 1 -­‐ .7881 = 0.2119 Part II. Mrs. Pugh went on vacation and could not be reached. Before she left, she turned in her individual student grades to her principal. The parent of the student who made a 68 in the class called and insisted to know the class average of her child’s class by the end of the day. Unfortunately, the principal could not retrieve the exact class average because he only had individual student scores, but he told the parent that he could give her a range of scores that the class average would most likely be located within by the end of the day. He first took a random sample of five students and calculated the average of the five students. He did this by using the random function on a calculator, choosing 5 numbers from 1-­‐50. The numbers he got are 43, 12, 24, 35, 5. 9. Locate these values on the dotplot. 1 corresponds to the lowest test score, and 50 corresponds to the highest test score. What are the associated test scores? 92, 72, 82, 89, 68 10. Find the average of these 5 test scores. 80.6 11. Is this sample mean the same as the population mean? No 12. Why or why not? It was close, but the small sample size, 5, we would not expect it to be exactly accurate III. Although using a sample size of 5 gave him a good idea what the class average was, he wanted to give the parent a smaller range in which the average could be located. He figured that Mrs. Pugh had at least 25 students in each of her classes. He took 50 random samples of size 25, calculated their means, and recorded them in a dotplot. Dot
Measures fromSample of Collection
1 Plot
77 78 79 80 81 82 83 84 85 86
mean25grades
Estimate the mean and standard deviation of this dotplot . 13. x = around 81.5 14. σ x =
σ
n
= 2.074 The principal wants to be 95% confident in the range of scores that he gives to the parent. Using the mean and standard deviation that you just approximated above for the class average of 25 students, answer the following, showing all work: 15. He called the parent and told her that he was very confident that the class average was between _______________ and _______________. 77.4 and 85.6 Part IV. The parent wanted to know how this semester’s grades compared to the previous semester’s grades. Here is a list of grades for Mrs. Pugh’s Advanced Algebra students from the previous semester (she only had 48 students that semester). 65,66,68,70,70,72,72,72,73,73,75,75,75,77,77,78,78,78,80,80,80,80,81,81,82,82,83,83,84,84,84,85,85,8
6,88,88,89,90,90,91,92,92,93,93,94,94,96,97 What is the mean and standard deviation of the distribution of individual grades? 16. x = _______________ 81.69 17. σ = _______________ 8.26 18. What could the principal tell the parent? Use at least 4 statistics to justify your answer. Answers will vary. Examples of statistics to use: Mean is the same, Standard deviation is smaller, Range is 32 compared to 40, mode is 80 compared to 90, median is 81.5 compared to 82, IQR is 13.75 compared to 16.5. Some reference should be made to the fact the mean is the same but the scores were less spread out the previous semester.