AP Statistics. Study Guide

Transcription

1 Measuring Relative Standing Standardized Values and z-scores AP Statistics Percentiles Rank the data lowest to highest. Counting up from the lowest value to the select data point we discover the percentile that is represented by the data. For Example: Suppose that you received a 78 on a 90 point quiz. You are ranked 30 th out of the 50 individual who took the test. Your score is: % 50 = = or in the 60th percentile. 40% of the students ranked higher than you. Another definition is that For any particular number p between 0 and 100, the pth percentile is a value such the p percent of the observations in the data set fall at or below that value. Introduction to Statistics & Data Analysis Peck, Olsen, and Devore Thomson (2008) ISBN 10: Page 1

2 Density Curves 1. Always plot your data: make a graph, usually a histogram or stemplot. 2. Look for the overall pattern (shape, center, spread) and for striking deviations such as outliers. 3. Calculate a numerical summary to briefly describe center (median or mean), spread (range, IQR, standard deviation) 4. Sometimes the overall patter of a large number of observations is so regular that we can describe it by a smooth curve. Doing so can help us describe the location of individual observations with a distribution. Page 2

3 Normal Curve Bell shaped density curve with several key properties: A density curve that is: Symmetric Single peaked Bell shaped Describes a Normal distributions Empirical Rule ( ) Remember these are percentages percentages are a proportion! Page 3

4 See Example 2.9 for a detailed explanation and step-by-step process. Page 4

5 Be prepared to use Table A or the calculator. The area under the curve is always between 0 and 1. The values expressed in Table A can also be expressed as a percentage or proportion. Normal Probability Plot CAUTION: When you examine the Normal probability plot, look for shapes that show clear departures from Normality (linear plot). Don t overreact to minor wiggles in the plot. Sample Problems with Key Try to answer the problem first then use the key. 1. What two population parameters determine the shape of a normal curve? (They make a normal curve tall and skinny or short and fat.) A. median and mean B. mode and standard deviation C. median and standard deviation D. mean and mode E. mean and standard deviation Page 5

6 2. Suppose a population of individuals has a mean weight of 160 pounds, with a population standard deviation of 30 pounds. According to the empirical rule, what percent of the population would be between 100 and 220 pounds? A. 10% B. 68% C. 95% D. 99.7% E. None of the above 3. Given N(544, 103) What is the approximate percentage of scares between 500 and 700? A. 50 B. 60 C. 70 D. 80 E Given N(544, 103) what is the approximate percentage of applicants who scored above a 450? A. 82 B. 82 C. 26 D. 11 E. 1 Page 6

7 5. Given N(544, 103) find the score at the upper quartile A. 1 B. 475 C. 470 D. 0 E To the nearest whole number, what percentile is associated with z=-.68? A. 10th percentile B. 40th percentile C. 50th percentile D. 25th percentile E. 75th percentile 7. A z-score is called a standardized score because you can: A. translate any x-value into a z-score. B. translate any x-value from a normal distribution into a z-score. C. translate z-scores into a proportion, a percentile, or a probability of the normal curve. D. use z-scores to find the area between a z-score and the mean, or the area below a z- score. E. use them to compare x-values! to a universal standard, in this case, the standard normal distribution. Page 7

8 Use the following information to answer Questions 8 and 9: Runner s World reports that the times of the finishers in the New York City 10-km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes. 8. Find the proportion of runners who finish in less than 43 minutes. 9. Find the proportion of runners who take more than 70 minutes to finish. 10. True or False: In a normal distribution, the mean, median, and mode all have the same value and the graph of the distribution is symmetric. 11. In terms of std deviations, where are the inflection points in a normal curve? A. 1 std deviation left AND 1 std deviation right of the mean B. 1 std deviation left AND 2 std deviations right of the mean C. 2 std deviations left AND 2 std deviations right of the mean D. At the mean and median value E. Halfway between the mean and the two most extreme outliers. Use the following information for Questions 12 and 13: A population of bolts has a mean thickness of 20 mm, with a population standard deviation of.01 mm. Page 8

9 12. Give, in mm, a min and max thickness that includes 68% of the population of bolts. A to mm B to mm C to mm D to mm E to mm 13. Give in mm, a min and max thickness that will include 95% of the population of bolts. A to mm B to mm C to mm D to mm E. These can t be accurately computed 14. Using the empirical rule, you can assume that what percent of the normal distribution is outside two standard deviations of the mean in either direction? A. 50% B. 10% C. 5% D. 1% E. Can t be decided Page 9

10 15. To the nearest whole number, what percentile is associated with z = + 1.2? A. 25th percentile B. 50th percentile C.75th percentile D.88th percentile E. 12th percentile 16. What area, to the nearest whole percent, of the normal curve is located between z = -0.6 and z = +1.4? A. 64% B. 91% C. 27% D. 50% E. 95% 17. What two population parameters determine the shape of a normal curve? (They make a normal curve tall and skinny or short and fat.) A. median and mean B. mode and standard deviation C. median and standard deviation D. mean and mode E. mean and standard deviation 18. Suppose a population of individuals has a mean weight of 160 pounds, with a population standard deviation of 30 pounds. According to the empirical rule, what percent of the population would be between 100 and 220 pounds? Page 10

11 A. 10% B. 68% C. 95% D. 99.7% E. None of the above 19. Assume that normal curve A and normal curve B have identical population means. Assume further than A has a greater population std. Deviation than B. Which curve is taller, and why? A. Curve A is taller because it has fewer inflection points B. Curve A is taller because smaller std. Deviations produce wider curves. C. Curve B is taller because its median is greater. D. Curve B is taller because smaller std. Deviations produce thinner curves. E. The curves are the same height Page 11