Linear Regression Models Directions: Use the information below to solve the problems in this packet. Packets are due at the end of the period and students who do not finish will be required to come in for lunch and make it up or be assigned Saturday School. Once you have finished this packet, be sure to check your folder for the AM Exercise the goes with this material. Properties of the Correlation Coefficient, r 1. - 1 r 1 2. When r > 0, there is a positive linear correlation. 3. When r < 0, there is a negative linear correlation. 4. When r 1, there is a strong linear correlation. 5. When r 0, there is weak or no linear correlation. *This means, that if I have an r value near positive or negative one, the model is a good fit for the situation. To calculate the correlation coefficient, use the instructions attached and find the r value given. Linear Correlation **Read page 159 of your textbook for an example of how to use a linear regression models to predict future values. This will help you answers questions on the next page.**
1. The numbers of insured commercial banks y (in thousands) in the United States for the years 1987 to 1996 are shown in the table. (Source: Federal Deposit Insurance Corporation). Year (x) # of insured banks (y) 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 (0) (1) (2) (3) (4) (5) (6) (7) (8) (9) 13.70 13.12 12.71 12.34 11.92 11.46 10.96 10.45 9.94 9.53 Make a scatterplot of the data, letting x represent the number of years since 1987. a) Use a graphing calculator to fit a linear function to the data. (see your instruction sheet) b) Compute the correlation coefficient. r = Is this line a good fit?. c) What does this statistic mean concerning the relationship between year and number of insured commercial banks? (Fill in sentence frame) The relationship between year and number of insured commercial banks has a, linear correlation. (use pictures from notes and the scatterplot you create.) d) What would be the slope and y-intercept for a regression line based on this data? The slope would be and the y-intercept would be, because if you calculate the linear regression model using a calculator, you get the equation. e) Graph the function of best fit with the scatterplot of the data. (see Instruction sheet) f) With the function found in part (c), predict the average number of insured commercial banks in 2000 and 2005. (See example in textbook)
2) The accompanying table shows the percent of the adult population that married before age 25 in several different years. Using the year as the independent variable, find the linear regression equation. Round the regression coefficients to the nearest hundredth. Using the equation found above, estimate the percent of the adult population in the year 2009 that will marry before age 25, and round to the nearest tenth of a percent. 3) Two different tests were designed to measure understanding of a topic. The two tests were given to ten students with the following results: a) Write an equation for the line of best fit (round slope and intercept to the nearest hundredth). b) Predict the score, to the nearest integer, on test y for a student who scored 87 on test x.
5) Since 1990, fireworks usage nationwide has grown, as shown in the accompanying table, where t represents the number of years since 1990, and p represents the fireworks usage per year, in millions of pounds. a) Find the equation of the linear regression model for this set of data, where t is the independent variable. Round values to four decimal places. b) Using this equation, determine in what year fireworks usage would have reached 99 million pounds. c) Based on this linear model, how many millions of pounds of fireworks would be used in the year 2014? Round your answer to the nearest tenth.
6) The accompanying table illustrates the number of movie theaters showing a popular film and the film's weekly gross earnings, in millions of dollars. a) Write the linear regression equation for this set of data, rounding values to five decimal places. b) Using this linear regression equation, find the approximate gross earnings, in millions of dollars, generated by 610 theaters. Round your answer to two decimal places. c) Find the minimum number of theaters that would generate at least 7.65 million dollars in gross earnings in one week.