1 Descriptive Statistics

Transcription

1 1 Descriptive Statistics 1. Below is a list of test scores for a small class: 100, 98, 97, 94, 100, 90, 4 (a) What is the average test score x? (b) What is the median test score m? 2. Bill Gates, the founder of Microsoft, is worth approximately 65 billion dollars. If Bill is in a room with 99 other people, each of whom have no money at all, what is the average net worth of the people in the room? 3. This question refers to the following data 12, 19, 28, 36, 41, 50, 59, 65, 67, 72, 83 (a) Give the five-number summary and box plot for this data set. (b) If a new observation is added to the data set, what is the maximum amount by which the new median will differ from the old median? (c) If a new observation is added to the data set, what is the maximum amount by which the new first quartile will differ from the old first quartile? (d) If a new observation is added to the data set, what is the maximum amount by which the new third quartile will differ from the old third quartile? (e) Compute the standard deviation for the data set. 4. This question refers to the box plot below 1

2 (a) Is the data represented by this box plot skewed left or skewed right? (b) Which do you think is larger for this data: the median or the mean? (c) Estimate the five-number summary for the data represented by this box plot. 5. This question uses this data set: Cloud data (Cleveland, Kleiner, and Tukey. (1983). Graphical Methods for Data Analysis. Wadsworth International Group, Belmont, CA, 351) 2

3 Unseeded Clouds Seeded Clouds The mean value for seeded clouds is , and the mean value for unseeded clouds is (a) If an additional observation of a seeded cloud gives 337 acre-feet of rainfall, what effect will this have on the mean number of acre-feet for seeded clouds? (b) If an additional observation of an unseeded cloud gives 881 acrefeet of rainfall, what effect will this have on the mean number of acre-feet for unseeded clouds? (c) The five-number summaries for each variable are as follows: Seeded Unseeded Min Q m Q Max Use the 1.5 IQR rule to classify outliers in the data above. (Recall that this rule classifies an observation as an outlier if it lies more than 1.5 IQR below the first quartile or more than 1.5 IQR above the third quartile.) 3

4 (d) If an additional observation of an unseeded cloud gives 1011 acre feet of rainfall, what effect will this have on mean and the median? On which quantity will the effect be greater? (e) Calculate the deviation and squared deviation for the observation in the unseeded data. (f) Calculate the deviation and squared deviation for the observation 430 in the seeded data (g) The standard deviations are for the seeded data and for the unseeded data. If an additional observation of a seeded cloud gives 1,881 acre feet of rainfall, what effect will this have on the standard deviation for the seeded data? (h) How many observations of seeded clouds are more than two standard deviations away from the mean? Answer the same question for unseeded clouds. 6. The data below are the percentages of high school graduates from each of the fifty states in the year 2000 who took the SAT test. Calculate the mean, standard deviation, and five number summary for this data. Use the 1.5 IQR rule to identify any outliers. Does the data look symmetric, skewed, or neither? Suppose that the histogram below is for a data set that consists of n = 100 lengths of salmon (in inches) sampled from a river in Oregon. 4

5 20 Fictional salmon lengths (a) What percentage of the fish in the sample were less than 10 inches long? (b) How many fish in the sample were more than 15 inches long? (c) How long was the longest fish in the sample? (d) Do you think that m > x or m < x? 8. Suppose that the histogram below is for a data set that consists of a sample of weights of golden retrievers (in pounds) Fictional golden retriever weights

6 (a) How many dogs were in the sample? (b) What percentage of the dogs in the sample weighed more than 100 pounds? (c) What percentage of the dogs in the sample weighed between 40 and 60 pounds? (d) Estimate the mean of the data set. 9. The chart below is a relative frequency histogram (a) If there were n = 140 observations in the sample, about how many of the observations are more than 20? (b) What fraction of the observations are between 18 and 20? (c) If you picked one of these observations from the sample at random, what is the probability that it would be less than 18? 6