Name Mr. Ressman Room #222

Transcription

1 Name Mr. Ressman Room #222

2 Basic Definitions: Notes 3.1 1) Statistics 2) Qualitative 3) Quantitative 4) Biased 5) Unbiased Classify each item as a qualitative descriptor or a quantitative descriptor. a. Scores on the unit 8 test b. Opinions of students on the pep rally c. The blood pressure of each student in class d. The nationality of each student in school

3 What are the Measures of Central Tendency? 1. Mean -- Ex. #1: Find the mean of: 93, 82, 65, 74, Median Ex. #2: Find the median of: 17, 18, 17, 19, 20, 46, 17 Ex. #3: Find the median of: 1, 9, 7, 3, 5, 6 3. Mode Ex. #4: Find the mode of: 17, 17, 17, 18, 19, 20, 21, 46 Ex. #5: Find the mode of: 8, 8, 9, 10, 11, 11, 12 Ex. #6: Find the mode of: 2, 5, 7, 9 4. Range Ex. #7: Find the range: Calculator steps to find the Mean (x ) and Median:

4 Practice Problems! P 3.1 Show all necessary work in solving each problem. 1. Rosario and Enrique are in the same math class. On the first five tests, Rosario received scores of: 78, 77, 64, 86, and 70. Enrique received scores of 90, 61, 79, 73, and 87. How much higher was Enrique s average than Rosario s average? a. 2 points b. 75 points c. 78 points d. 3 points THINK! PLAN Is the question simply asking you to find the average? What two things will you need to find to answer this question? a. b. CALCULATE Be sure to show what you are entering into your calculator. Label each answer. DID YOU ANSWER THE QUESTION YET??? How will you answer the question? Calculate below and circle your answer. 2. What was the median high temperature in Middletown during the 7-day period shown in the table? THINK! What is the 1 st step to finding the median? SHOW WORK (list # s, cross off) Match your answer to the correct answer.

5 3. Sara s test scores in math were: 64, 80, 88, 78, 60, 92, 84, 76, 86, 78, 72, and 90. Determine the mean, median, mode, and range of Sara s test scores. THINK! How many answers are you finding? How many test scores are listed? Find the MEAN (show work): Find the MEDIAN (show work, be sure to double check that your list has the same number of scores that you identified above!): Find the MODE: Find the RANGE: 4. From January 3 rd to January 7 th, Buffalo recorded the following daily high temperatures: 5, 7, 6, 5, and 7. Which statement about the temperatures is true? a. mean < median b. mean = mode c. median = mode d. mean = median THINK: What data will you need to find BEFORE selecting an answer? List it below and calculate each. PLAN: How will you use your data to answer the question? Do you know what the symbol in choice A means? *HINT* Plug in the numbers and see which statement is TRUE!!!

6 Notes 3.2 BOX AND WHISKER GRAPHS Data can be represented in graphs which quickly give information about the scores or data. Definitions Quartile 1 (25 th percentile) Quartile 2 (median) Quartile 3 (75 th percentile) Minimum Maximum Interquartile Range (IQR) 1. Draw a line down the middle of the hearts. 2. Draw a line down the middle of the bottom group of hearts. 3. Draw a line down the middle of the top group of hearts No matter the size, each QUARTILE contains % To create a Box and Whisker plot, you need the 5-number summary MIN, Quartile 1, MEDIAN, Quartile 3, MAX

7 Ex. 1 Based on the box and whisker graph above, find each of the following: a) Median b) Lower Quartile c) 75 th Percentile d) Range e) IQR Steps to make a box and whisker graph by hand: 1. Arrange the data in order 2. Find the median (quartile 2) a) find the median of the lower ½ (quartile 1) b) find the median of the upper ½ (quartile 3) 3. Plot dots at the minimum, maximum, Q1, Q2, Q3 4. Draw a box around Q1 and Q3, draw vertical lines through Q2 5. Draw whiskers to the minimum and maximum 6. The interquartile range (IQR) is Q3 Q1 Ex. 2 Draw a box and whisker plot for: 12, 13, 15, 15, 18, 22, 24, 26 What is the IQR?

8 Calculator Work Notes 3.3 We can use the same calculator steps for finding Mean and Median to find the rest of the information needed for a box and whisker graph! Step 1: Enter as a new document under Lists and Spreadsheets. Step 2: Perform a One-variable Stat calculation (MENU 4 1 1) Step 3: Then add a new page using: ctrl doc (Choice 5: Data and Statistics) Step 4: Click your variable on the x-axis Step 5: Menu - 1: Plot Type 2: Box Plot It will look like this. then Ex. 3 Enter the following data into your calculator: 48, 84, 27, 91, 18, 48, 28, 81, 78, 73, 84, 12, 48, 45, 21 Record the following after performing a Stat Calculation: MinX Q1X MedianX Q3X MaxX Create a Box and Whisker plot on your calculator and show ME! Sketch that Box and Whisker plot below.

9 Ex. 4 Now perform all the same calculations on this new set of data. Be sure to show me your finished product. Lengths of giant centipedes (in cm): 9, 14, 10, 8, 22, 15, 11, 21, 27 MinX Q1X MedianX Q3X MaxX Sketch your box and whisker graph below, and make sure it is to scale! P 3.3 1) Fifteen students took a 20-question quiz and received the following scores: 5, 6, 7, 8, 9, 9, 9, 10, 10, 14, 17, 17, 18, 19, 19 Find: MinX Q1X MedianX Q3X MaxX Create a box and whisker of your data: 2) Jaquan looked at his quiz scores shown below for the first and second quarters of his Algebra class. Semester 1: 78, 91, 88, 83, 94 Semester 2: 91, 96, 80, 77, 88, 85, 92 Which statement below about Jaquan s performance is correct? a) The interquartile range for quarter 1 is greater than the interquartile range for quarter 2. b) The median score for quarter 1 is greater than the median score for quarter 2. c) The mean score for quarter 2 is greater than the mean score for quarter 1. d) The third quartile for quarter 2 is greater than the third quartile for quarter 1.

10 Circle the number of the correct choice for each of the questions below (these are former Regents questions!) 3) 4) 5) 6)

11 Other Options? Notes 3.4 Box and Whisker graphs are just one way we can visually interpret data. We will now look at a few other ways. 1) Dot Plots A plot of each data value on a scale or number line Can we find our measures of central tendency with a dot plot? How? 2) Histograms A graph of data that groups the data based on intervals and represents the data in each interval by a bar. What are the pros and cons of using a histogram?

12 We can use the same calculator steps for finding Mean and Median and box and whisker graphs to illustrate either a dot plot or a histogram! Step 1: Enter as a new document under Lists and Spreadsheets. Step 2: Perform a One-variable Stat calculation (MENU 4 1 1) Step 3: Then add a new page using: ctrl doc (Choice 5: Data and Statistics) Step 4: Click your variable on the x-axis Step 5: Menu - 1: Plot Type 2: Dot Plot or Histogram Ex. 1) Use the following data to construct both a Dot Plot and a Histogram. Compare your graphs with your neighbors. Be sure to show me at least one of the two graphs. Fifteen students took a 20-question quiz and received the following scores: 5, 6, 7, 8, 9, 9, 9, 10, 10, 14, 17, 17, 18, 19, Skewed vs. Symmetrical Notes 3.5 When we view data on a graph (any type), we may notice things about how the data lines up. Not all data graphs have a nice, symmetrical shape. Here we will discuss what it means for data to be Skewed and what it looks like. Box and Whisker Plots: Box plots can often provide information about the shape of a data set. The examples below show some common patterns. Each of the above box plots illustrates a different skewness pattern. If most of the observations are concentrated on the low end of the scale, the distribution is skewed right; and vice versa. If a distribution is symmetric, the observations will be evenly split at the median, as shown above in the middle figure.

13 Frequency graphs can also be described as having a skewness pattern: In graph A, the data is concentrated at the low end (left) so it is considered to be skewed. (Note: the mean is than the median) In graph B, the data is concentrated at the high end (right) so it is considered to be skewed. (Note: the mean is than the median) In graph C, the data is equally distributed around the middle so it is considered. (Note: the mean is the median) Normal Curves Although we don t do much with normal curves this year, we can view the skewness in much the same way.

14 Ex. 1 Describe each distribution. The distribution is shown as a histogram and as a box and whisker plot. For distribution b, describe how the mean is related to the median. Ex. 2 Answer the questions that accompany each graph to understand the story behind the data.

15 P 3.5 1) Sam said that a typical flight delay for the sixty BigAir Flights was approximately one hour. Do you agree? Why or why not? 2) Sam said that 50% of the twenty-two juniors at River City High School who participated in the walkathon walked at least ten miles. Do you agree? Who or why not? 3) Sam said that young people from the ages of 0 to 10 years old make up nearly one-third of the Kenyan population. Do you agree? Why or why not?

16 Variation Within a Data Set Notes 3.6 Measures of central tendency give us numbers that describe the typical data value in each data set. But, they do not let us know how much variation there in in the data set. Two data sets can have the same mean but look radically different depending on how varied the numbers are in the set. Ex. 1 The two data sets below each have equal means but differ in the variation within the data set. Use your calculator to determine the interquartile range (IQR) of each data set. The IQR is defined as the difference between the third quartile value and the first quartile value. Set #1: 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11 Set #2: 5, 5, 6, 6, 7, 7, 8, 8, 9, 9 The interquartile range gives a good measure of how spread out the data set is. But, the best measure of variation within a data set is the standard deviation. The actual calculation of standard deviation is complex and we will not go into it here. We will rely on our calculators for its calculation. Ex. 2 Using the same data sets above, use your calculator to produce the standard deviation (shown as σ x on the calculator) of the two data sets. Round your answers to the nearest tenth. Data Set #1: Data Set #2: Standard Deviation The standard deviation of a data set tells us, on average, how far a data point is away from the mean of the data set. The larger the standard deviation, the greater the variation within the data set. Ex. 3 A farm is studying the weight of baby chickens (chicks) after 1 week of growth. They find the weight, in ounces, of 20 chicks. The weights are shown below. Find the mean, the IQR, and the standard deviation for this data set. Round any non-integer values to the nearest tenth. Include appropriate units in your answers. 2, 1, 3, 4, 2, 2, 3, 1, 5, 3, 4, 4, 5, 6, 3, 8, 5, 4, 6, 3 Mean = IQR = Standard Deviation (σ x ) = Ex. 4 Without using your calculator, determine which of the following data sets would have a standard deviation closest to zero? Be prepared to defend your answer! a) {-5, -2, -1, 0 1, 2, 5} b) {5, 8, 10, 16, 20} c) {11, 11, 12, 13, 13} d) {3, 7, 11, 11, 11, 18}

17 Ex. 5 A marketing company is trying to determine how much diversity there is in the age of people who drink different soft drinks. They take a sample of people and ask them which soda they prefer. For the two sodas, the age of those people who preferred them is given below. Gushing Grape: 18, 16, 22, 16, 28, 18, 21, 38, 22, 29, 25, 44, 36, 27, 40 Killer Kola: 25, 22, 18, 30, 27, 19, 22, 28, 25, 19, 23, 29, 26, 18, 20 a) Explain why a standard deviation is a better measure of the diversity in age than the mean. b) Which soda appears to have a greater diversity in the age of people who prefer it? How did you decide that? c) Use your calculator to determine the sample standard deviation, normally given as sx, for both data sets. Round your answers to the nearest tenth. Did this answer reinforce your pick from (b)? How? Population Versus Sample Standard Deviation When we are working with every possible data point of interest, we call this a population and use the population standard deviation, σ x. When we have only a sample of all possible values we use the sample standard deviation, sx. The formulas for these two differ slightly, so their values tend to be slightly different.

18 Outliers Notes 3.7 What is an outlier? Any piece of data (high or low) that does not fit with the bulk of the data. Ex. 1 Examine the following ages for a recent concert. 18, 17, 20, 16, 9, 22, 18, 21, 19, 73 What do you notice? Let s put them in order 9, 16, 17, 18, 18, 19, 20, 21, 22, 73 Notice how the end pieces of data do not seem to fit with the others. We call these outliers. Ex. 2 Baseball 2012: The table below shows runs batted in (RBI s) for select players on the Arizona Diamondbacks baseball team for a) Which player appears to be an outlier for RBI data? b) Verify your possible outlier choice by preparing a box and whisker plot on your calculator. Fill in the following values: Q1 Median Q3 c) Find the mean for this data set. Mean = d) Now, REMOVE the OUTLIER and find: Mean Median e) Which value (mean or median) appears to remain more stable, whether the outlier is included or not?

19 Notes 3.8 Univariate and Bivariate Statistics Univariate Bivariate Identify each example below as using either univariate (U) or bivariate (B) statistics: Ex. 1 A study was conducted to see how the number of hours studying effects test scores Ex. 2 A study was conducted to determine the heights of the players on the basketball team Ex. 3 A study was done to determine the number of miles driven and amount of gas used Ex. 4 Research was conducted to determine the number of McDonald s in New York State Correlation If there is a correlation, the dots appear to make a Positively sloped line Negatively sloped line

20 Thinking about Linear Relationships Below are three scatter plots. Each one represents a data set with eight observations. The scales on the x and y-axes have been left off these plots on purpose so you will have to think carefully about the relationships displayed. 1. If one of these scatter plots represents the relationship between height and weight for 8 adults, which scatter plot do you think it is and why? 2. If one of these scatter plots represents the relationship between height and SAT math scores for 8 high school seniors, which scatter plot do you think it is and why? 3. If one of these scatter plots represents the relationship between the weight of a car and fuel efficiency for 8 cars, which scatter plot do you think it is and why? 4. Which of these scatter plots does not appear to represent a linear relationship? Explain the reason you selected this scatter plot.

21 P 3.8 Now, try to analyze some of the scatter plots on your own and draw conclusions. 1) You are traveling around the US with friends. After spending a day in a town that is 2,000 feet above sea level, you plan to spend the next several days in a town that is 5,000 feet above sea level. Is this town likely to have more or fewer clear days per year than the town that is 2,000 feet above sea level? Explain your answer. 2) You plan to buy a bike helmet. Based on data presented in this lesson, will buying the most expensive bike helmet give you a helmet with the highest quality rating? Explain your answer.

22 Line of Best Fit Notes 3.9 What is it? To create the equation of the line of best fit, you need two pieces of information: a. b. What can a line of best fit be used for? Interpolate Extrapolate Ex. 1 a) What kind of correlation is there between hours of sleep and test scores? b) Draw a line of best fit. c) What is the y-intercept of your line? What is the slope of your line? (Pay attention to the scale on the y-axis) d) Write the equation of your line of best fit. e) Use your line of best fit to estimate your test score if you got 11 hours of sleep. f) If this interpolating or extrapolating? Why?

23 To determine if a line is a good fit for the data, we look at the correlation coefficient. * A positive correlation will have a number from 0 to 1. Any number.8 or larger is considered a strong positive correlation. * A negative correlation will have a number from 0 to -1. Any number -.8 or less is considered a strong negative correlation. The closer a correlation is to 1 or -1, the more closely the data resembles a line.

24 Notes 3.10 Calculator Keystrokes for Line of Best Fit To find your line of best fit, there are two ways. First enter your data in a list and create a scatter plot. Method #1 while still on the scatter plot screen a. MENU b. #4 Analyze c. #6 Regression d. #1 Show linear (mx + b) - this method draws the line of best fit for you and gives you the equation Method #2 Go to a fresh calculator page. a. #6 Statistics b. #1 Stat Calculations c. #3 Linear Regression (mx + b) d. Choose your x variable and y variable then hit OK - this method gives you the slope and y-intercept which you will need to create an equation from - it also gives you the correlation coefficient (r value) Ex. 2 The table below shows the weight of an alligator at various times during a feeding trial. a. Make a scatter plot of the data on your calculator. b. Perform a linear regression. Write the equation of the line of best fit to the nearest hundredth. Slope = y-intercept = equation of line of best fit: c. What does the slope represent? d. Use your equation to predict the weight of this alligator at week 52. Did you interpolate or extrapolate? e. Graph your line on your scatter plot. Sketch your graph. f. What is the value of your correlation coefficient (r) for your line of best fit? g. This shows a correlation.

25 P ) The table below shows the hours of training and the finishing time in minutes of eight runners. a. Make a scatter plot of the data on your calculator. b. Perform a linear regression. Write the equation of the line of best fit to the nearest hundredth. Slope = y-intercept = equation of line of best fit: c. What does the slope represent? d. Use your equation to predict the weight of this alligator at week 52. Did you interpolate or extrapolate? e. Graph your line on your scatter plot. Sketch your graph below. f. What is the value of your correlation coefficient (r) for your line of best fit? g. This shows a correlation.

26 Residuals Residual = Observed y-value - Predicted y-value A residual is the difference between the observed y-value (from scatter plot) and the predicted y-value (from regression equation line). It is the vertical distance from the actual plotted point to the point on the regression line. You can think of a residual as how far the data "fall" from the regression line (sometimes referred to as "observed error"). Linear associations are the most popular statistical relationships since they are easy to read and interpret. We will spend most our time working with linear relationships, and residuals can tell us when we have an appropriate linear model. When you look at your scatter plot, and you are unsure if the shape (curve) you chose for your regression equation will create the best model, a residual plot will help you decide as to whether the model you chose will, or will not, be an appropriate linear model. A residual plot is a scatter plot that shows the residuals on the vertical axis and the independent variable on the horizontal axis. The plot will help you to decide on whether a linear model is appropriate for your data. Appropriate linear model: when plots are randomly placed, above and below x-axis (y = 0). Appropriate non-linear model: when plots follow a pattern, resembling a curve.

27 Regents Review Questions on Linear Regression and Correlation Coefficient

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30 Review for Unit Test 1. Gary Golfer is in the Thursday Night Men s Amateur League. Which teammate has the least consistent scores? [1] Mean = 38, Standard Dev = 4.1 [3] Mean = 39, Standard Dev = 3.8 [2] Mean = 36, Standard Dev = 1.6 [4] Mean = 38, Standard Dev = Which of the following r-values would best match the scatter plot below? [1] r = [2] r = [3] r = 0.86 [4] r = Which of the following data sets is not bivariate? [1] The ages of a track team and the number of minutes it takes each member to run a mile. [2] The average number of hours students spend doing homework each night. [3] The number of hours a student spends studying for a test and the grade received on the test. [4] The number of years of schooling a person has and their current income. 4. The box-and-whisker plot shown below represents the prices of tickets sold, in hundreds, at MetLife Stadium in New Jersey. What conclusion can be made using this plot? [1] The median price is 600 dollars. [2] The mean price is 400 dollars. [3] The range of prices is 300 to 600 dollars. [4] 50% of the prices were between 300 & 600 dollars.

31 5. Based on the line of best fit drawn, what resting heart rate would you expect a person who exercises exactly 4 hours per week to have? [1] 48 [2] 63 [3] 80 [4] Assume you want to determine the most popular sports team in America. Which of the following sampling techniques would provide the most unbiased sample? [1] Asking all the people in one section at Ralph Wilson Stadium, Home of the Bills [2] Asking every tenth person who enters Ralph Wilson Stadium [3] Asking all the members of a local gym [4] Randomly selecting 50 names from the latest US Census 7. A statistics teacher does a project with each of her 4 classes, comparing the average temperature each day as it relates to the percentage of kids who have their homework done the next day. Listed below are the results for each of the four classes, stating their correlation coefficient. Which of the following classes showed the weakest correlation between average temperature and percentage of homework done? [1] Class A (0.64) [3] Class C (0.96) [2] Class B (0.81) [4] Class D (0.98)

32 8. Isaiah collects data from two different companies, each with four employees. The results of the study, based on each worker s age and salary are listed in the tables below. Company 1 Company 2 Age Salary Age Salary 25 30, , , , , , , ,000 Which statement about these data is false? [1] The mean salaries in both companies are greater than $36,000. [2] The median salary in company 1 is greater than the mean salary in company 2. [3] The salary range in company 2 is greater than the salary range in company 1. [4] The mean age of workers is the same at both companies. 9. A random sample of n = 1000 portfolio rate returns (measured as a percentage) is taken; here is the data: The distribution of the rate of returns is best described as: [1] Skewed Left [3] Symmetric [2] Skewed Right [4] Uniform

33 10. A local ice cream store s profit rises when the weather outside gets warmer. (a) Based on this information, does the relationship between the change in temperature and the change in profits show a positive or negative correlation? (b) Is the relationship above also causal? Justify your answer. 11. During the school year, there are 4 quarter grades and the final exam. If Tucker needs exactly an average of 65% to pass the year and his 4 quarter grades are 68, 60, 55, and 70, what is the lowest grade Tucker can earn on his final exam to pass the year? Show how you arrived at your answer. 12. The following diagram represents the total wins each season the Boston Red Sox had over the last 18 years. (a) What percentage of the seasons totals are less than 72 wins per season? (b) What is the interquartile range?

34 13. The prices of seven race cars sold last week are listed below. Which measure of central tendency, mean or median, best represents the value of the seven race cars. Justify your answer. Price per Race Car Number of Race Cars $126,000 1 $140,000 2 $180,000 1 $400,000 2 $819, A large industrial plant studies the relationship between the number of hours devoted to safety training and the number of work hours lost due to workplace accidents. Work Time Lost Due to Accidents (hours) Time Spent on Safety Training (hours) (a) Draw a reasonable line of best fit on the given scatter plot. (b) Using your line from (a) estimate the work time lost due to accidents for an individual with 70 hours of safety training. (c) Is part (b), interpolating or extrapolating? Explain.

35 15. The following table shows the height in inches of 10 students on the tennis team, as well as their weight in pounds. Height, inches (x) Weight, lbs (y) (a) Find the equation for the line of best fit. Round the m and b values to 3 decimal place accuracy. (b) If Max is 190 pounds, using your line of best fit from part a, how tall, to the nearest inch, is Max? Show your work. (c) What is the correlation coefficient, rounded to the nearest hundredth? (d) Explain what the correlation coefficient suggests in the context of the problem.