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Learning to do Linear Regressions Exploring Data on the TI-84 The average height in inches of United States children from ages 7 to 15 is given: Age (yrs.) |7 |8 |9 |10 |11 |12 |13 |14 |15 | |Height (in.) |46.97 |48.43 |52.00 |53.98 |55.98 |57.99 |60.00 |62.01 |63.86 | |to enter the data STAT, 1:Edit, enter data into L1 and L2 1. to set up the plot of the data 2nd ,STAT PLOT, 1:PLOT1, ENTER On Type: scatter Xlist: L1 Ylist: L2 Mark (any) 2. to graph the scatter plot ZOOM , 9: ZoomStat 3. to turn on correlation coefficient CATALOG DiagnosticON ENTER 4. to find the linear regression STAT, CALC, 4:LinReg(ax+b) L1, L2, y1 and paste it into the y1 (found under VARS,Y-VARS,1:Function, 1:Y1) equation on the home screen Analysis: 1. What is the slope of the line' (Round answer 3 places behind decimal and include units.) In a complete sentence, explain the real-world meaning of this slope. (In discussion round to nearest whole number.) 2. What is the y-intercept' (Round answer 3 places behind decimal and include units.) In a complete sentence, explain the real world meaning for the y intercept. (In discussion round to nearest whole number.) Is this reasonable' 3. What is a reasonable domain and range' Explain why you chose these and include units. 4. Interpolate to find the average height of a 9½-year-old child. (Express answer in feet and inches.) Is it feasible to use this model to find this answer' Explain. 5. Extrapolate to find the height of a 25 year old. (Express answer in feet and inches rounding to the nearest ½ inch.) Is it feasible to use this model to find this answer' Explain. 6. Use the model to determine the average age of a child (in years and months) who is 55 in. tall. Is it feasible to use this model to find this answer' Explain. 7. Discuss the correlation coefficient and if this is a good regression. Extension: Add 16, 17, and 18 years old to L1 and 64 inches every year. Find new regression equation and paste in y2. Discuss how the outliers affect the equation and the validity of this new model. Learning to do Linear Regressions Answer Key [pic] [pic] [pic] Analysis: (1) The slope of the line is 2.138 in/yr., which means that an average US child grows approximately 2 inches in one year. (2) The y-intercept is 32.168 inches, which means when an average baby is born, his/her length is 32 inches. This is high since most babies are around 20 - 25 inches when born. (3) A reasonable domain is 0 years to 20 years because this is the time of life when you are growing. A reasonable range is 8 inches to 72 inches because premature babies may be this length and average people are below 6 feet may be this height. (Answers may vary.) (4) The average height of a 9½ year-old child is 4'4½". This is reasonable because it is between the heights at 9 and 10 years old. (5) The height of a 35 year old would be 8'11". This is not feasible because you stop growing at some point in time. (6) The average age of a child who is 55 inches tall is 10 years and 8 months. This is feasible because the height is between 9 and 10 years old. Extension: The child stops growing and the outliers affect the model making it unreliable for the ages 7 – 15. Only 94% of the points lie in a small band about the regression line as opposed for 99% in the first equation. Three Mathematical Models for Lines Notes for Teacher Best Fit Line by hand – “guesstimate” 1. The line should show the direction of the points. (The smallest rectangle that contains the points shows the direction.) 2. The line should divide the points equally (as many above the line as below). 3. As many points as possible should be one the line (after satisfying #1 and #2). 4. The points above the line (or below) should not be concentrated at one end. Median-Median Line – three points are selected to represent the entire data set and the equation that best fits these three points is taken as the best fit line for the entire set of data. (This lines is called a resistant line because it is not influenced by outliers.) 1. Order the data by the domain values. 2. Divide the data into three equal groups. If not divisible by thee, then split them so that the first and last group is the same size. 3. Order the y-values in each group 4. Choose the representative point of each group, which will be the median point in each group. (If odd #, choose the middle point. If even #, average the 2 middle values.) 5. Find the equation of the line through the representative points in groups 1 and 3. 6. Find the equation of the line parallel to the first line through the median point in the 2nd group. 7. Find the equation of the line one-third of the way closest to the line formed by groups 1 and 2. (Find the average of the y-intercepts using the y intercept formed by the line of groups 1 and 3 twice and y-intercept of group 2’s line once.) This is the median-median line. Least Squares Line – the line that the graphing calculator uses. To find the coefficients of the linear equation y = ax + b use the following formulas: [pic] Correlation Coefficient - r 1. r is the strength of the linear regression. –1 < r < 1. ( is a perfect correlation between x and y. Zero is no linear correlation. The sign refers to ( slope. [pic] 2. r2 is the measure of the fraction of total variation in the values of y or how closely the data points are clustered in a small band. (i.e. if r = .5 then r2 = .25 or 25%) Other Terminology 1. Interpolate – Estimating a value between those given in a table of data 2. Extrapolate – Use the model to estimate values beyond the present range of values. 3. Outlier – A value in a data set that is uncharacteristic of most of the data. Louisiana Mathematics Grade-Level Expectations for 9th grade (Benchmarks) Linear Regressions Number and Relations 4.  Distinguish between an exact and an approximate answer, and recognize errors introduced by the use of approximate numbers with technology (N-3-H) (N-4-H) (N-7-H) Algebra 9. Model real-life situations using linear expressions, equations, and inequalities (A-1-H) (D-2-H) (P-5-H) 10. Identify independent and dependent variables in real-life relationships (A-1-H) 13. Translate between the characteristics defining a line (i.e., slope, intercepts, points) and both its equation and graph (A-2-H) (G-3-H) 15. Translate among tabular, graphical, and algebraic representations of functions and real-life situations (A-3-H) (P-1-H) (P-2-H) Geometry 25. Explain slope as a representation of “rate of change” (G-3-H) (A-1-H) Data Analysis, Probability, and Discrete Math 29.    Create a scatter plot from a set of data and determine if the relationship is linear or nonlinear (D-1-H) (D-6-H) (D-7-H) Patterns, Relations, and Functions 36.    Identify the domain and range of functions (P-1-H) 37.    Analyze real-life relationships that can be modeled by linear functions (P-1-H) (P-5-H) 38.    Identify and describe the characteristics of families of linear functions, with and without technology (P-3-H) 39.    Compare and contrast linear functions algebraically in terms of their rates of change and intercepts (P-4-H) 40.    Explain how the graph of a linear function changes as the coefficients or constants are changed in the function’s symbolic representation (P-4-H)