Chapter 5: Statistical Reasoning Section 5.1 Chapter 5: Statistical Reasoning Section 5.1 Exploring Data Terminology: Mean: A measure of central tendency determined by dividing the sum of all the values in the set. Median: A measure of central tendency represented by the middle value of an ordered data set. Ex1. The median of the data set {3,6,4,9,5} would be 5. You must remember to reorder a list of data prior to determining the median so the list becomes {3,4,5,6,9} with the middle number of the set of data being 5 (not 4 as it appears to be in the original unordered list) Ex2. The median of a data set that has an even number of digits is calculated by adding the two middle values together and divide by two (ie find the value in between them). For the data set {7,12,20,17,5,21}, once we rearrange the data in ascending order {5,7,12,17,20,21}, there are two middle numbers 12 & 17. The median is 12+17 = 29 = 14.5. 2 2 Mode: A measure of central tendency represented by the value that occurs most often in a data set. Note: If a set of data has no repeating data values or more than one set that repeats more than others, the data is said to have no mode. Ex. For the data set {2,5,7,5,6,3,5,9,2}, the mode would be 5 as it occurs more than any other value. Range: The difference between the maximum and minimum values in a set of data. Ex. The range for the data set {2,5,7,16,22,27} would be 27 2 = 25. Outlier: A value in a data set that is very different from the other values in a set. Ex. In the data set {18,52,63,64,65,69,72,72,78,83,87,91}, 18 would be considered an outlier. Note: Not all sets of data have an outlier. Line Plot: A graph that records each data value in a data set as a point above a number line. 50 100 150 Dispersion: A measure that varies by the spread among the data in a set; dispersion has a value of zero if all the data in a set is identical, and it increases as the data becomes more spread. Range can be used to determine the size of dispersion. Large range = Large Dispersion.
Chapter 5: Statistical Reasoning Section 5.1 Using Measures of Central Tendency Ex1. Given the set of test scores from a Unit 1 Test below: 92 52 63 75 82 97 87 49 100 82 (a) Determine the measures of central tendency for the data (mean, median, and mode). (b) Determine the range of the data. (c) Is there any piece of data that would be considered an outlier? Why? (d) Does this data have a large dispersion or small dispersion. Explain your reasoning.
Chapter 5: Statistical Reasoning Section 5.1 Ex2. Given the set of test scores from a Unit 2 Test below: (a) Determine the measures of central tendency for the data (mean, median, and mode). (b) Determine the range of the data. (c) Is there any piece of data that would be considered an outlier? Why? (d) Does this data have a large dispersion or small dispersion. Explain your reasoning. (e) Did students do better on Unit 1 or Unit 2.
Chapter 5: Statistical Reasoning Section 5.2 Section 5.2: Histograms, Frequency Tables and Polygons Terminology: Frequency Distribution: A set of intervals (Table or graphs), usually of equal width, into which raw data is organized; each interval is associated with a frequency that indicates the number of measurements in this interval. Usually when determining the size of the intervals, we consider the highest and lowest values of data that we are dealing with and attempt to create intervals that can encompass this range. We aim to have an approximately 10 intervals. It is sometimes best to determine the range of your data and divide by 10 to determine the size of your intervals. Histogram: The graph of a frequency distribution, in which equal intervals of values are marked on a horizontal axis and the frequencies associated with these intervals are indicated by the rectangles drawn for these intervals. REMEMBER: A histogram, like all other graphs, must have a title, labeled axes, and a consistent scaling on each axis. Frequency Polygon: The graph of a frequency distribution, produced by joining the midpoints of the intervals using straight lines. Building a Histogram and Frequency Polygon The following data represents the flow rates of the Red River from 1950 to 1999, as recorded at the Redwood Bridge in Winnipeg, Manitoba. Year Flow Rate (m 3 /s) Maximum Water Flow Rates for the Red River 1950-1999 Year Flow Year Flow Year Flow Rate Rate Rate (m 3 /s) (m 3 /s) (m 3 /s) Year Flow Rate (m 3 /s) 1950 3058 1960 1965 1970 2280 1980 881 1990 396 1951 1065 1961 481 1971 1526 1981 159 1991 280 1952 1008 1962 1688 1972 1589 1982 1458 1992 1399 1953 357 1963 660 1973 530 1983 1393 1993 946 1954 524 1964 1002 1974 2718 1984 1048 1994 1121 1955 1521 1965 1809 1975 1671 1985 991 1995 1877 1956 1974 1966 2498 1976 1807 1986 1812 1996 3058 1957 654 1967 1727 1977 187 1987 2339 1997 4587 1958 524 1968 510 1978 1750 1988 564 1998 1557 1959 991 1969 2209 1979 3030 1989 1390 1999 2180
Chapter 5: Statistical Reasoning Section 5.2 Ex1. Construct a frequency table and a histogram to represent the Red River Flow rate data. Interpret your results to determine the most common water flow rate. Frequency Table: Flow Rate (The intervals - Includes the lower limit but not the upper) Tally (A tally of the number of data points that are in each interval) Frequency (The number of items in the interval as a number) Midpoint of Range (Used For Frequency Polygon) Histogram:
Chapter 5: Statistical Reasoning Section 5.2 Frequency Polygon: Interpretation of Data:
Chapter 5: Statistical Reasoning Section 5.2 Ex2: The amount of money withdrawn from a local ATM machine is recorded for a single Wednesday as shown: 20 120 50 70 60 80 140 120 80 160 40 100 120 80 60 110 100 100 80 180 160 40 100 50 180 200 50 80 200 140 160 60 40 80 60 140 110 20 160 100 140 160 200 140 20 80 20 100 70 150 70 100 40 20 120 40 140 100 40 50 180 60 40 200 80 Display the data in a frequency table, histogram, and frequency polygon. (For this example you are permitted to draw your polygon on the same diagram as the histogram) Frequency Table: Amount Withdrawn Tally Frequency Midpoint of Range Histogram:
Chapter 5: Statistical Reasoning Section 5.2 Interpreting a Histogram The magnitude of an earthquake is measured using the Richter scale. Examine the histograms for the frequency of earthquake magnitudes in Canada from 2005 to 2009. Which of these years could have had the most damage sustained from earthquakes? Solution:
Chapter 5: Statistical Reasoning Section 5.3 Section 5.3: Standard Deviation Terminology: Deviation: The difference between a data value and the mean for the same set of data. Standard Deviation: A measure of the dispersion or scatter of data values in relation to the mean. A low standard deviation indicates that most data values are close to the mean, and a high standard deviation indicates that most data values are scattered farther from the mean. Formulae for Standard Deviation: Mean: x = x n Standard Deviation: σ = (x x)2 n
Chapter 5: Statistical Reasoning Section 5.3 Calculating Standard Deviation for a Set of Data Brenda works part time in the canteen at his local community centre. One of his tasks is to unload delivery trucks. He wondered about the accuracy of the mass measurements given on two cartons that contained sunflower seeds. He decided to measure the masses of the 20 bags in the two cartons. (a) The first carton contained 227 g bags of sunflower seeds. The measurements of their masses are showed below: Masses of the 227 g bags (g) 228 220 233 227 230 227 221 229 224 235 224 231 226 232 218 218 229 232 236 223 Determine the standard deviation for the mass of the bags. Data Value (x) Difference (x x) Difference Squared (x x) Total
Chapter 5: Statistical Reasoning Section 5.3 (b) The first carton contained 454 g bags of sunflower seeds. The measurements of their masses are showed below: Masses of the 454 g bags (g) 458 445 457 458 452 457 445 452 463 455 451 460 455 453 456 459 451 455 456 450 Determine the standard deviation for the mass of the bags. Data Value (x) Difference (x x) Difference Squared (x x) Total
Chapter 5: Statistical Reasoning Section 5.3 Ex2. Given below are the list of test scores on last year s Math 1201 Final Exam. Determine the standard deviation of the class. What does this tell you about the dispersion of the marks. 56 93 48 73 81 88 65 52 32 21 45 59 60 62 85 100 90 61 50 43 41 34 80 54 Data Value (x) Difference (x x) Difference Squared (x x) Total (x): x = Total ((x x) 2 ): (x x) 2 = Mean (x): Standard Deviation (σ): x = x = n σ = (x x)2 = n
Chapter 5: Statistical Reasoning Section 5.4 Section 5.4: Normal Distribution Terminology: Normal Curve: A symmetrical curve that represents the normal distribution, also known as the bell curve. Normal Distribution: Data that, when graphed as a histogram or frequency polygon, results in a unimodal symmetric distribution about the mean. In a normal distribution all three measures of central tendency are equal (or exceptionally close). Normal Distribution and Standard Deviation The normal distribution has a special relationship with standard deviation. That is that when data complies to a normal distribution (ie, makes a bell curve) the data is always distributed such that: 68% of Data falls within 1 Standard Deviations (1 SD) of the Mean 95% of Data falls within 2 Standard Deviations (2 SD) of the Mean 99.7% of Data falls within 3 Standard Deviations (3 SD) of the Mean 0.3% often represents the outliers in a set of data.
Chapter 5: Statistical Reasoning Section 5.4 The normal distribution of data is shown below. Examining the Properties of a Normal Distribution Ex1. Heidi is opening a new snowboard shop near a local resort. She knows that the recommended length of a snowboard is related to a person s height. Her research ski Height Frequency (in) 61 or 3 shorter 61-62 4 62-63 10 63-64 18 64-65 30 65-66 52 66-67 64 67-68 116 68-69 128 69-70 147 70-71 129 shows that most of the snowboarders that visit the resort are males, 20-39 years old. To ensure that she stocks the most popular snowboard lengths, she collects
Chapter 5: Statistical Reasoning Section 5.4 height data from 1000 Canadian man, 20 to 39 years old. Construct a histogram to display the data. Does the data resemble a normal distribution? If so, determine the standard deviation and use it to determine the most common sizes of snow boards Heidi should purchase. Analyzing a Normal Distribution Ex. Jim raises Siberian Husky sled dogs at his kennel. He knows, from the data he has collected over the years, that the weights of adult dogs are normally distributed, with a mean of 52.5 lbs. and a standard deviation of 2.4 lbs. Use this information to sketch a normal distribution curve. Use your curve to determine what percentage of dogs at Jim s kennel would you expect to have a weight between 47.7 lbs and 54.9 lbs.
Chapter 5: Statistical Reasoning Section 5.4 Ex. Sally packages fruit at a local convenience store. She knows that the mean weight of the fruit packages is 254 g. She has determined that the standard deviation for the fruit packages is 8.7 g. Use this information to sketch a normal distribution and determine the percentage of fruit packages that are between 236.6 g and 280.1 g. Comparing Normally Distributed Data Ex. Two baseball teams flew to the North American Indigenous Games. The members of each team had carry-on luggage for their sports equipment. The masses of the carry-on luggage were normally distributed, with the characteristics shown below: Team μ (kg) σ (kg) Men 6.35 1.04 Women 6.35 0.59 (a) Which team s luggage was closest in weights? How do you know?
Chapter 5: Statistical Reasoning Section 5.4 (b) When graphed on the same graph, how do you think they will differ in appearance? (c) Sketch both on the same graph (d) If the women s team won at the Games and each play received a metal and a souvenir weighing 1.2 kg. How would the graph of their return weight differ from the original? Determining if Data is a Normal Distribution For data to accurately be considered a Normal Distribution, it must satisfy the following characteristics: Mean and Median (and Mode) must be equal or very close and data should be close to normally distributed. (1 SD = 68%; 2 SD = 95%; 3 SD = 99.7%). To determine if the data is normally distributed you will need the mean, median, standard deviation, and a histogram (or frequency polygon) Ex. Mandy wants to buy a new cellphone. She researches the cellphone she is considering and finds the following data on its longevity (ie phone life), in years 2.0 2.4 3.3 1.7 2.5 3.7 2.0 2.3 2.9 2.2 2.3 2.7 2.5 2.7 1.9 2.4 2.6 2.7 2.8 2.5 1.7 1.1 3.1 3.2 3.1 2.9 2.9 3.0 2.1 2.6 2.6 2.2 2.7 1.8 2.4 2.5 2.4 2.3 2.5 2.6 3.2 2.1 3.4 2.2 2.7 1.9 2.9 2.6 2.7 2.8
Chapter 5: Statistical Reasoning Section 5.4 (a) Does the data approximate a normal distribution? (Hint use SD to determine Bins) Bin Tally Frequency (b) If Mandy bought this cellphone, what is the likelihood that it will last for more than three years? Ex. Wallace is organizing his movie collection. He decides to record the length of each movie, in minutes. 91 129 95 96 96 90 101 87 100 90 86 78 105 99 81 106 101 122 91 102 89 125 162 155 89 89 180 94 84 99 73 100 99 100 117 135 100 89 87 110 125 103 94 99 98 102 96 88 154 144 (a) Does the data approximate a normal distribution? (Hint use SD to determine Bins) Bin Tally Frequency
Chapter 5: Statistical Reasoning Section 5.4 Section 5.5: Z-Scores Terminology: Z-Score A standardized value that indicates the number of standard deviations of a data value above or below the mean. Standard Normal Distribution A normal distribution that has a mean of zero and a standard deviation of one. Z-Score Table A table that displays the fraction of data with a z-score that is less than any given data value in a standard normal distribution. There is a z-score table on pages 580-581 of your text book.
Chapter 5: Statistical Reasoning Section 5.5 Solving for the z results in a formula gives us the z- Ex. Brianna score: and Charles belong to a x μ running club in z = Vancouver. σ Part of their training involves a 200 m sprint. Below are normally distributed times for the 200 m sprint in Vancouver and on a recent trip to Lake Louise. At higher altitudes, run times improve. Location Calculating Z-Score: For any given score, x, from a normal distribution: Altitude (m) Club Mean Time (μ) for 200 m run (s) x = μ + zσ where z represents the number of standard deviations of the score from the mean. Club Standard Deviation (σ) for run (s) Brianna's Run Time (s) Charles' Run Time (s) Vancouver 4 25.75 0.62 24.95 25.45 Lake Louise 1661 25.57 0.60 24.77 26.24 (a) Determine at which location Brianna's Run time was better when compared to the club's overall performance.
Chapter 5: Statistical Reasoning Section 5.5 (b) Determine at which location Charles' run time was better when compared to the club's overall performance Using Z-Scores to Determine Percent of Data Below a Value Ex. IQ tests are sometimes used to measures a person's intellectual capacity at a particular time. IQ scores are normally distributed, with a mean of 100 and a standard deviation of 15. If a person scores 119 on an IQ test, how does this compare with the scores of the general population? Ex. If Gudie scored a 95 on her IQ test, how does this compare with the general population?
Chapter 5: Statistical Reasoning Section 5.5 Using Z-Score to Determine Data Values Ex. Athletes should replace their running shoes before the shoe loses their ability to absorb shock. Running shoes lose their shock absorption after a mean distance of 640 km, with a standard deviation of 160 km. Zack is an elite runner and wants to replace his shoes after a distance when only 25% of people would replace their shoes. At what distance should he replace his shoes?
Chapter 5: Statistical Reasoning Section 5.5 Ex. Quinn is a recreational runner. She plans to replace her shoes when 70% of people would replace their shoes. After how many kilometers should she replace her shoes? Quality Control Problems Ex. The ABC Company produces bungee cords. When the manufacturing process is running well, the length of the bungee cords produced are normally distributed, with a mean of 45.2 cm and a standard deviation of 1.3 cm. Bungee cords that are shorter than 42.0 cm or longer than 48.0 cm are rejected by the quality control workers. (a) If 20 000 bungee cords are manufactured each day, how many bungee cords are expected to be rejected by the quality control workers?
Chapter 5: Statistical Reasoning Section 5.5 (b) What action might the company take as a result of these findings? Ex. A hardwood flooring company produces flooring that has an average thickness of 175 mm, with a standard deviation of 0.4 mm. For premium-quality floors, the flooring must have a thickness between 174 mm and 175.6 mm. (a) What percent, to the nearest whole number of the total production can be sold for premium-quality floors?
Chapter 5: Statistical Reasoning Section 5.5 (b) If on a randomly selected day, 30 000 pieces of flooring are created, how many pieces of flooring are rejected for premium-flooring? Determining Warranty Periods Ex. A manufacturer of personal music players has determined that the mean life of the device is 32.4 months, with a standard deviation of 6.3 months. What length of warranty should be offered if the manufacturer wants to restrict repairs to less than 1.5% of all the players sold?
Chapter 5: Statistical Reasoning Section 5.5 Ex. After some research a smart phone company finds that their product has a mean lifetime of 21.3 months with a standard deviation of 7.1 months. What warranty should the company offer if they wish to restrict repairs to less than 2.6% of all the smart phones they sell. Section 5.6: Confidence Intervals Terminology: Margin of Error The possible difference between the estimate of the value you re trying to determine, as determined from a random sample, and the true value for the population; the margin of error is generally expressed as a plus or minus percentage, such as ±5%. Confidence Interval The interval in which the true value you re trying to determine is estimated to lie, with a stated degree of probability; the confidence interval may be expressed using ± notation, such as 54.0% ± 3.5%, or as a range, such as 50.5% to 57.5% Confidence Level The likelihood that the result for the true population lies within the range of the confidence interval; surveys and other studies usually use a confidence level of 95% although 90% or 99% percent is sometimes used. Analyzing and Applying Survey Results
Chapter 5: Statistical Reasoning Section 5.6 Ex. A telephone survey of 600 randomly selected people was conducted in an urban area with a population of 92 500 people. The survey determined that 76% of people, from 18 to 34 years of age, have an account with social media. The results are accurate within plus or minus 4 percent points, 19 times out of 20. Calculate the range of people that have a social media account, and determine the certainty of the result. Ex. The same telephone survey was conducted by a different company, using a sample size of 600 people selected from both urban and rural areas. The total population for this area was 98 000 people. According to the survey, 76% of people, ages 18 to 34 have an account with social media. These results are accurate within ±5.3%, 99 times out of 100. Calculate the range of people that have a social media account, and determine the certainty of the result.
Chapter 5: Statistical Reasoning Section 5.6 Analyzing the Effect of Sample Size on Margin of Error and Confidence Intervals Ex. Polling organizations in Canada frequently survey samples of the population to gauge voter preference prior to an election. People are asked: 1. If an election were held today, which party would you vote for? If they say they don t know, then they are asked: 2. Which party are you leaning towards voting for? The results of three different polls taken during the first week of November 2010 are shown below. The results of each poll are considered accurate 19 times out of 20. Polling Organizati on & Data Concervative (%) Liberal (%) NDP (%) Bloc Quebecois (%) Green Party (%) Undecided (%) EKOS 29 29 19 9 11 12.6 Sample Size: 1815 Margin of Error: ±2.3% NANOS 37 32 15 11 5 19.2 Sample Size: 844 Margin of Error: ±3.4% IPSOS 35 29 12 11 12 n/a Sample Size: 1000 Margin of Error: ±3.1% How does the sample size used in a poll affect:
Chapter 5: Statistical Reasoning Section 5.6 (a) The margin of error in the reported poll? (b) The confidence Interval in the reported poll? Analyzing the Effect of Confidence Levels on Sample Size Ex. To meet regulation standards, baseballs must have a mass from 142.0 g to 149.0 g. A manufacturing company has set its production equipment to create baseballs that have a mean mass of 145.0 g. To ensure that the production equipment continues to operate as expected, the quality control engineer takes a random sample of baseballs each day and measures their mass to determine the mean mass. If the mean mass of the random sample is 144.7 g to 145.3 g, then the production equipment is running correctly. If the mean mass of the sample is outside the acceptable level, the production equipment is shut down and adjusted. The quality control engineer refers to the chart shown to the right when conducting random sampling. Confidence Level Sample Size Needed 99% 110 95% 65 90% 45 (a) What is the confidence interval and margin of error the engineer is using for these quality control tests.
Chapter 5: Statistical Reasoning Section 5.6 (b) Interpret the table (c) What is the relationship between confidence level and the sample size Analyzing Statistical Data to Support a Position Ex. A poll was conducted to ask voters the following question: If an election were held today, whom would you vote for? The results indicated that 53% would vote for Jackson and 47% would vote for Kennedy. The results were stated as being accurate within 3.8 percentage points, 19 times out of 20. Who will win the election? Ex. For the upcoming school election, students were polled to see which student body president candidate was most likely to win the election. The results showed that 45% of
Chapter 5: Statistical Reasoning Section 5.6 students will likely vote for Smith and 55% of students will vote for Rice. The results were stated as being accurate with ±6.2%, 9 times out of 10. The school newspaper predicts that Rice is guaranteed to win the election, is this prediction sound (valid)?