822 CHAPTER 12 Statistics Do you believe that there is a relationship between the time a person studies for an exam and the exam grade the person receives? Is there a relationship between the age of a car and the value of the car? Can we predict the value of a car based on the age of the car? In this section, we will learn how to determine whether there is a relationship between two quantities and, if so, how strong that relationship is. We will also learn how to determine the equation of the line that best describes the relationship between two quantities. Linear Correlation and Regression SECTION 12.6 LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand linear correlation. 7 Calculate a linear correlation coefficient. 7 Understand linear regression. 7 Calculate the equation of the line of best fit and use the equation to make estimations or predictions. Why This Is Important There are many real-life applications in which a relationship exists between two variables, such as the amount of time spent studying for an exam and the exam score. Once a linear relationship between the amount of time spent studying for an exam score and the exam score is determined, we can use that relationship to predict an exam score based on the amount of time spent studying. determine the values that correspond to 1.1, 1.5, 2.0, and 2.5 standard deviations below the mean. 8.63, 9.83, 11.33, 12.83; 2.03, 0.83, − − 0.67, 2.17, respectively d) By observing the 30 pieces of data, determine the actual percent of quiz scores between ± ± ± ± 1.1 standard deviations from the mean. 1.5 standard deviations from the mean. 2 standard deviations from the mean. 2.5 standard deviations from the mean. ≈ ≈ 56.7, 93.3%, 100%, 100% e) Place the percents determined in part (d) in the third row of the chart. ≈ ≈ 56.7, 93.3%, 100%, 100% f) Compare the percents in the third row of the chart with the minimum percents in the first row and the normal percents in the second row, and then make a judgment as to whether this set of 30 scores is approximately normally distributed. No 95. Determine a value of z such that ≤ z 0 and 11.9% of the standard normal curve lies to the left of the z-value. −1.18 Recreational Mathematics 96. Z-Scores Ask your instructor for the class mean and class standard deviation for one of the exams taken by your class. For that exam, calculate the z-score for your exam grade. How many standard deviations is your exam grade away from the mean? Answers will vary. 97. Standard Deviation If the mean score on a math quiz is 12.0 and 77% of the students in your class scored between 9.6 and 14.4, determine the standard deviation of the quiz scores. 2 Research Activity 98. Collect A Set of Data In this project, you actually become the statistician. a) Select a project of interest to you in which data must be collected. b) Write a proposal and submit it to your instructor for approval. In the proposal, discuss the aims of your project and how you plan to gather the data to make your sample unbiased. c) After your proposal has been approved, gather 50 pieces of data by the method you proposed. d) Rank the data from smallest to largest. e) Compute the mean, median, mode, and midrange of the data. f) Determine the range and standard deviation of the data. You may round the mean to the nearest tenth when computing the standard deviation. g) Construct a frequency distribution, histogram, frequency polygon, and stem-and-leaf display of your data. Select your first class so that there will be between 5 and 12 classes. Be sure to label your histogram and frequency polygon. h) Does your distribution appear to be normal? Explain your answer. Does it appear to be another type of distribution discussed? Explain. i) Determine whether your distribution is approximately normal by using the technique discussed in Exercise 94. Ulrich Mueller/Shutterstock
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