SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide - Chapter Statistics. Sampling Techniques
Statistics Statistics is the art and science of gathering, analyzing, and making inferences from numerical information (data) obtained in an experiment. Statistics are divided into two main braches. Descriptive statistics is concerned with the collection, organization, and analysis of data. Inferential statistics is concerned with the making of generalizations or predictions of the data collected. Slide - Statisticians A statistician s interest lies in drawing conclusions about possible outcomes through observations of only a few particular events. The population consists of all items or people of interest. The sample includes some of the items in the population. When a statistician draws a conclusion from a sample, there is always the possibility that the conclusion is incorrect. Slide -5 Types of Sampling A random sampling occurs if a sample is drawn in such a way that each time an item is selected, each item has an equal chance of being drawn. When a sample is obtained by drawing every nth item on a list or production line, the sample is a systematic sample. A cluster sample is referred to as an area sample because it is applied on a geographical basis. Slide -6
Types of Sampling continued Stratified sampling involves dividing the population by characteristics such as gender, race, religion, or income. Convenience sampling uses data that is easily obtained and can be extremely biased. Slide -7 Example: Identifying Sampling Techniques A raffle ticket is drawn by a blindfolded person at a festival to win a grand prize. Students at an elementary are classified according to their present grade level. Then, a random sample of three students from each grade are chosen to represent their class. Every sixth car on highway is stopped for a vehicle inspection. Slide -8 Example: Identifying Sampling Techniques continued Voters are classified based on their polling location. A random sample of four polling locations are selected. All the voters from the precinct are included in the sample. The first people entering a water park are asked if they are wearing sunscreen. Solution: a) Random d) Cluster b) Stratified e) Convenience c) Systematic Slide -9
. The Misuses of Statistics Misuses of Statistics Many individuals, businesses, and advertising firms misuse statistics to their own advantage. When examining statistical information consider the following: Was the sample used to gather the statistical data unbiased and of sufficient size? Is the statistical statement ambiguous, could it be interpreted in more than one way? Slide - Example: Misleading Statistics An advertisement says, Fly Speedway Airlines and Save %. Here there is not enough information given. The Save % could be off the original ticket price, the ticket price when you buy two tickets or of another airline s ticket price. A helped wanted ad read, Salesperson wanted for Ryan s Furniture Store. Average Salary: $,. The word average can be very misleading. If most of the salespeople earn $, to $5, and the owner earns $76,, this average salary is not a fair representation. Slide -
Charts and Graphs Charts and graphs can also be misleading. Even though the data is displayed correctly, adjusting the vertical scale of a graph can give a different impression. A circle graph can be misleading if the sum of the parts of the graphs do not add up to %. Slide - Example: Misleading Graphs While each graph presents identical information, the vertical scales have been altered. Sales Dollars (in thousands) 75 5 5 75 5 5 Sales 99 Years Dollars (in thousands) 5 99 Years Slide -. Frequency Distributions 5
Example The number of pets per family is recorded for families surveyed. Construct a frequency distribution of the following data: Slide -6 Solution Number of Pets Frequency 6 8 Slide -7 Rules for Data Grouped by Classes The classes should be of the same width. The classes should not overlap. Each piece of data should belong to only one class. Slide -8 6
Definitions Classes 5 9 Lower class limits Upper class limits 5 9 5 9 Midpoint of a class is found by adding the lower and upper class limits and dividing the sum by. Slide -9 Example The following set of data represents the distance, in miles, 5 randomly selected second grade students live from school. 6.8 5. 9.7.8 8.7.5 5.9.8 5.7..8 9.6.5 7.. Construct a frequency distribution with the first class. Slide - Solution First, rearrange the data from lowest to highest. # of miles from school Frequency..5.8-5..8 5.9 8.7.5 5. 6.8 9.6.8 5.7 7. 9.7. -.. - 6. 6. - 8. 8. -. 5 Slide - 7
. Statistical Graphs Circle Graphs Circle graphs (also known as pie charts) are often used to compare parts of one or more components of the whole to the whole. Slide - Example According to a recent hospital survey of patients the following table indicates how often hospitals used four different kinds of painkillers. Use the information to construct a circle graph illustrating the percent each painkiller was used. Aspirin Ibuprofen Acetaminophen Other 56 6 Slide - 8
Solution Determine the measure of the corresponding central angle. Painkiller Number of Patients Percent of Total Measure of Central Angle Aspirin 56 = 56 8%.8 6 =.8 Ibuprofen = 5%.5 6 = 87. Acetaminophen 6 = 6 8%.8 6 = 8.8 Other = %. 6 =. Total % 6 Slide -5 Solution continued Use a protractor to construct a circle graph and label it properly. Hospital Painkiller Use Ibuprofen 5% Aspirin 8% Other Acetaminophe Slide -6 Histogram A histogram is a graph with observed values on its horizontal scale and frequencies on it vertical scale. Example: Construct a histogram of the frequency distribution. # of pets Frequency 6 8 Slide -7 9
Solution Number of Pets per Family Frequency 8 6 Number of Pets # of pets Frequency 6 8 Slide -8 Frequency Polygon Number of Pets per Family Frequency 8 6 Number of Pets Slide -9 Stem-and-Leaf Display A stem-and-leaf display is a tool that organizes and groups the data while allowing us to see the actual values that make up the data. The left group of digits is called the stem. The right group of digits is called the leaf. Slide -
Example The table below indicates the number of miles workers have to drive to work. construct a stem-and-leaf display. 8 8 5 5 7 7 6 6 5 9 Slide - Solution Data 8 8 5 5 7 7 6 6 5 9 89 5678 567 5 Slide -.5 Measures of Central Tendency
Definitions An average is a number that is representative of a group of data. The arithmetic mean, or simply the mean is symbolized by or by the Greek letter mu, µ. x Slide - Mean The mean, x is the sum of the data divided by the number of pieces of data. The formula for calculating the mean is x x = n x where represents the sum of all the data and n represents the number of pieces of data. Slide -5 Example-find the mean Find the mean amount of money parents spent on new school supplies and clothes if 5 parents randomly surveyed replied as follows: $7 $65 $67 $5 $ x x $7 + $65 + $67 + $5 + $ = = n 5 $8 = = $68.8 5 Slide -6
Median The median is the value in the middle of a set of ranked data. Example: Determine the mean of $7 $65 $67 $5 $. Rank the data from smallest to largest. $5 $ $7 $65 $67 middle value (median) Slide -7 Example: Median (even data) Determine the median of the following set of data: 8, 5, 9,,, 7,,, 6,. Rank the data: 6 7 8 9 5 There are pieces of data so the median will lie halfway between the two middle pieces the 7 and 8. The median is (7 + 8)/ = 7.5 6 7 8 9 5 Slide -8 Mode The mode is the piece of data that occurs most frequently. Example: Determine the mode of the data set:,,, 6, 7, 8, 9,,, 5. The mode is since is occurs twice and the other values only occur once. Slide -9
Midrange The midrange is the value halfway between the lowest (L) and highest (H) values in a set of data. lowest value + highest value Midrange = Example: Find the midrange of the data set $7, $65, $67, $5, $. $5 + $67 Midrange = = $ Slide - Example The weights of eight Labrador retrievers rounded to the nearest pound are 85, 9, 88, 75, 9, 88, 8, and. Determine the a) mean b) median c) mode d) midrange e) rank the measures of central tendency from lowest to highest. Slide - Example--dog weights 85, 9, 88, 75, 9, 88, 8, Mean 85 + 9 + 88 + 75 + 9 + 88 + 8 + x = 8 77 = = 88.75 8 Median-rank the data 75, 8, 85, 88, 88, 9, 9, The median is 88. Slide -
Example--dog weights 85, 9, 88, 75, 9, 88, 8, Mode-the number that occurs most frequently. The mode is 88. Midrange = (L + H)/ = (75 + )/ = 88 Rank the measures 88.75, 88, 88, 88 Slide - Measures of Position Measures of position are often used to make comparisons. Two measures of position are percentiles and quartiles. Slide - To Find the Quartiles of a Set of Data Order the data from smallest to largest. Find the median, or nd quartile, of the set of data. If there are an odd number of pieces of data, the median is the middle value. If there are an even number of pieces of data, the median will be halfway between the two middle pieces of data. Slide -5 5
To Find the Quartiles of a Set of Data continued The first quartile, Q, is the median of the lower half of the data; that is, Q, is the median of the data less than Q. The third quartile, Q, is the median of the upper half of the data; that is, Q is the median of the data greater than Q. Slide -6 Example: Quartiles The weekly grocery bills for families are as follows. Determine Q, Q, and Q. 7 7 7 8 8 7 7 5 5 5 5 75 6 7 8 95 7 9 Slide -7 Example: Quartiles continued Order the data: 5 75 7 8 8 95 6 7 7 7 9 5 5 5 7 7 7 8 Q is the median of the entire data set which is 9. Q is the median of the numbers from 5 to 7 which is 95. Q is the median of the numbers from to which is 7. Slide -8 6
.6 Measures of Dispersion Measures of Dispersion Measures of dispersion are used to indicate the spread of the data. The range is the difference between the highest and lowest values; it indicates the total spread of the data. Slide -5 Example: Range Nine different employees were selected and the amount of their salary was recorded. Find the range of the salaries. $, $, $6,5 $56, $8, $7, $8,5 $,5 $56,75 Range = $56,75 $, = $,75 Slide -5 7
Standard Deviation The standard deviation measures how much the data differ from the mean. s = ( x x) n Slide -5 To Find the Standard Deviation of a Set of Data. Find the mean of the set of data.. Make a chart having three columns: Data Data Mean (Data Mean). List the data vertically under the column marked Data.. Subtract the mean from each piece of data and place the difference in the Data Mean column. Slide -5 To Find the Standard Deviation of a Set of Data continued 5. Square the values obtained in the Data Mean column and record these values in the (Data Mean) column. 6. Determine the sum of the values in the (Data Mean) column. 7. Divide the sum obtained in step 6 by n, where n is the number of pieces of data. 8. Determine the square root of the number obtained in step 7. This number is the standard deviation of the set of data. Slide -5 8
Example Find the standard deviation of the following prices of selected washing machines: $8, $7, $665, $68, $99, $99 Find the mean. x 665 + 7 + 68 + 8 + 99 + 99 8 x = = = = 5 n 6 6 Slide -55 Example continued, mean = 5 Data 7 8 99 665 68 99 Data Mean 97 5 5 7 5 (Data Mean) ( 97) = 88,9 5,756 6,5,8 8,9 8,65,56 Slide -56 Example continued, mean = 5 s =,56 6,56 s = = 9.5 5 The standard deviation is $9.5. Slide -57 9
.7 The Normal Curve Types of Distributions Rectangular Distribution J-shaped distribution Rectangular Distribution Frequency Values Slide -59 Types of Distributions continued Bimodal Skewed to right Slide -6
Types of Distributions continued Skewed to left Normal Slide -6 Normal Distribution In a normal distribution, the mean, median, and mode all have the same value. Z-scores determine how far, in terms of standard deviations, a given score is from the mean of the distribution. value of piece of data mean x µ z = = standard deviation σ Slide -6 Example: z-scores A normal distribution has a mean of 5 and a standard deviation of 5. Find z-scores for the following values. a) 55 b) 6 c) value of piece of data mean a) z = standard deviation 55 5 5 z55 = = = 5 5 A score of 55 is one standard deviation above the mean. Slide -6
Example: z-scores continued 6 5 b) z6 = = = 5 5 A score of 6 is standard deviations above the mean. 5 7 c) z = = =. 5 5 A score of is. standard deviations below the mean. Slide -6 To Find the Percent of Data Between any Two Values. Draw a diagram of the normal curve, indicating the area or percent to be determined.. Use the formula to convert the given values to z-scores. Indicate these z- scores on the diagram.. Look up the percent that corresponds to each z-score in Table. Slide -65 To Find the Percent of Data Between any Two Values continued. a) When finding the percent of data between two z-scores on the opposite side of the mean (when one z-score is positive and the other is negative), you find the sum of the individual percents. b) When finding the percent of data between two z-scores on the same side of the mean (when both z-scores are positive or both are negative), subtract the smaller percent from the larger percent. Slide -66
To Find the Percent of Data Between any Two Values continued c) When finding the percent of data to the right of a positive z-score or to the left of a negative z- score, subtract the percent of data between ) and z from 5%. d) When finding the percent of data to the left of a positive z-score or to the right of a negative z- score, add the percent of data between and z to 5%. Slide -67 Example Assume that the waiting times for customers at a popular restaurant before being seated for lunch at a popular restaurant before being seated for lunch are normally distributed with a mean of minutes and a standard deviation of min. a) Find the percent of customers who wait for at least minutes before being seated. b) Find the percent of customers who wait between 9 and 8 minutes before being seated. c) Find the percent of customers who wait at least 7 minutes before being seated. d) Find the percent of customers who wait less than 8 minutes before being seated. Slide -68 Solution wait for at least minutes Since minutes is the mean, half, or 5% of customers wait at least min before being seated. between 9 and 8 minutes 9 z9 = =. 8 z8 = =. Use table.7 page 8..% + 7.7% = 8.8% Slide -69
Solution continued at least 7 min less than 8 min 7 8 z7 = =.67 z8 = =. Use table.7 page 8. Use table.7 page 8. 5.% is between the mean and.8% is between the mean and.67... 5% 5.% =.7% 5%.8% = 9.% Thus,.7% of customers wait at Thus, 9.% of customers wait least 7 minutes. less than 8 minutes. Slide -7.8 Linear Correlation and Regression Linear Correlation Linear correlation is used to determine whether there is a relationship between two quantities and, if so, how strong the relationship is. The linear correlation coefficient, r, is a unitless measure that describes the strength of the linear relationship between two variables. If the value is positive, as one variable increases, the other increases. If the value is negative, as one variable increases, the other decreases. The variable, r, will always be a value between and inclusive. Slide -7
Scatter Diagrams A visual aid used with correlation is the scatter diagram, a plot of points (bivariate data). The independent variable, x, generally is a quantity that can be controlled. The dependant variable, y, is the other variable. The value of r is a measure of how far a set of points varies from a straight line. The greater the spread, the weaker the correlation and the closer the r value is to. Slide -7 Correlation Slide -7 Correlation Slide -75 5
Linear Correlation Coefficient The formula to calculate the correlation coefficient (r) is as follows: r = ( ) ( )( ) n xy x y ( ) ( ) ( ) ( ) n x x n y y Slide -76 Example: Words Per Minute versus Mistakes There are five applicants applying for a job as a medical transcriptionist. The following shows the results of the applicants when asked to type a chart. Determine the correlation coefficient between the words per minute typed and the number of mistakes. Applicant Ellen George Phillip Kendra Nancy Words per Minute 67 5 Mistakes 8 9 Slide -77 Solution We will call the words typed per minute, x, and the mistakes, y. List the values of x and y and calculate the necessary sums. WPM Mistakes x y x y xy 8 576 6 9 67 89 77 5 89 66 68 9 56 8 6 x = 9 y = 5 x =,7 y = 5 xy =,8 Slide -78 6
Solution continued The n in the formula represents the number of pieces of data. Here n = 5. r = n( xy) ( x)( y) n( x ) ( x) n( y ) ( y) 5( 8) ( 9)( 5) r = 5(,7) ( 9) 5( 5) ( 5) =,5,95 5(,7) 7,96 5( 5) 5 = 55 5,555 7,96 55 5 = 55.86 559 5 Slide -79 Solution continued Since.86 is fairly close to, there is a fairly strong positive correlation. This result implies that the more words typed per minute, the more mistakes made. Slide -8 Linear Regression Linear regression is the process of determining the linear relationship between two variables. The line of best fit (line of regression or the least square line) is the line such that the sum of the vertical distances from the line to the data points is a minimum. Slide -8 7
The Line of Best Fit Equation: y = mx + b, where ( ) ( )( ) ( ) ( ) ( ) n xy x y y m x m =, and b = n x x n Slide -8 Example Use the data in the previous example to find the equation of the line that relates the number of words per minute and the number of mistakes made while typing a chart. Graph the equation of the line of best fit on a scatter diagram that illustrates the set of bivariate points. Slide -8 Solution From the previous results, we know that ( ) ( )( ) n( x ) ( x) n xy x y m = 5(,8) (9)(5) m = 5(,7) 9 55 m = 559 m.8 Now we find the y-intercept, b. ( ) y m x b = n 5.8 9 b = 5.6 b = 6.5 5 ( ) Therefore the line of best fit is y =.8x + 6.5 Slide -8 8
Solution continued To graph y =.8x + 6.5, plot at least two points and draw the graph. x y 7.6 8.7 8.88 Slide -85 Solution continued Slide -86 9