Exam 2: Data

A regression analysis of company profits and the amount of money the company spent on advertising found $r^{2}=0.72$ . Which of these is true? I. The model can correctly predict the profit for 72% of companies. II. On average, about 72% of a company's profit results from advertising. III. On average, companies spend about 72% of their profits on advertising.

Another company's sales increase by the same percent each year. This growth is...

Students A growing school district tracks the student population growth over the years from 2008 to 2013. Here are the regression results and a residual plot. students =119.53+172.03 year Sample size: 6 R-sq =0.987 a. Explain why despite a high R-sq, this regression is not a successful model. To linearize the data, the log (base 10) was taken of the student population. Here are the results. Dependent Variable: $\log$ (students) Sample size: 6 $R-s q=0.994$ Parameter Estimate Std. Err. constant 2.871 0.0162 year 0.0389 0.00152 b. Describe the success of the linearization. c. Interpret R-sq in the context of this problem. d. Predict the student population in 2014.

Assembly line Your new job at Panasony is to do the final assembly of camcorders. As you learn how, you get faster. The company tells you that you will qualify for a raise if after 13 weeks your assembly time averages under 20 minutes. The data shows your average assembly time during each of your first 10 weeks. Week Time(min) 1 43 2 39 3 35 4 33 5 32 6 30 7 30 8 28 9 26 10 25 a. Which is the explanatory variable? b. What is the correlation between these variables? c. You want to predict whether or not you will qualify for that raise. Would it be appropriate to use a linear model? Explain.

Which is true? I. Random scatter in the residuals indicates a model with high predictive power. II. If two variables are very strongly associated, then the correlation between them will be near +1.0 or -1.0. III. The higher the correlation between two variables the more likely the association is based in cause and effect.

Music and grades (True Story) A couple of years ago a local newspaper published research results claiming a positive association between the number of years high school children had taken instrumental music lessons and their performances in school (GPA). a. What does "positive association" mean in this context? b. A group of parents then went to the School Board demanding more funding for music programs as a way to improve student chances for academic success in high school. As a statistician, do you agree or disagree with their reasoning? Explain briefly.

A study by a prominent psychologist found a moderately strong positive association between the number of hours of sleep a person gets and the person's ability to memorize information. a. Explain in the context of this problem what "positive association" means. b. Hoping to improve academic performance, the psychologist recommended the school board allow students to take a nap prior to any assessment. Discuss the psychologist's recommendations.

Shrimp From 1982 to 1990, there was a decrease in the number of white shrimp harvested from the Galveston Bay. Here is the regression analysis and a residual plot. The year has been shortened to two digits (82, 83……) and the dependent variable is the number of shrimp collected per hour. Dependent Variable: Shrimp/hour $\text { R-sq }=0.9496342$ s: 6.0232354 Parameter Estimate Std. Err. constant 816.71111 66.903419 year -8.9333333 0.77759635 a. Write the regression equation and define your variables. b. Find the correlation coefficient and interpret it in context. c. Interpret the value of the slope in context. d. In 1991, the shrimp production rebounded (in part due to the effects of El Nino) to 81 shrimp/hour. Find the value of this residual. e. The prediction for 1991 was very inaccurate. What name do statisticians give to this kind of prediction error? ^

Storks Data show that there is a positive association between the population of 17 European countries and the number of stork pairs in those countries. a. Briefly explain what "positive association" means in this context. b. Wildlife advocates want the stork population to grow, and jokingly suggest that citizens should be encourage to have children. As a statistician, what do you think of this plan? Explain briefly.

Associations For each pair of variables, indicate what association you expect: positive(+), negative(-), curved(C), or none (N). _ a. power level setting of a microwave; number of minutes it takes to boil water b. number of days it rained in a month (during the summer); number of times you mowed your lawn that month ___ c. number of hours a person has been up past a normal bedtime; number of minutes it takes the person to do a crossword puzzle ___ d. number of hockey games played in Minnesota during a week; sales of suntan lotion in Minnesota during that week ___ e. length of a student's hair; number of credits the student earned last year

All but one of the statements below contain a mistake. Which one could be true?

Identify what is wrong with each of the following statements: a. The correlation between Olympic gold medal times for the 800m hurdles and year is -0.66 seconds per year. b. The correlation between Olympic gold medal times for the 100m dash and year is -1.37. c. Since the correlation between Olympic gold medal times for the 800m hurdles and 100m dash is -0.41, the correlation between times for the 100m dash and the 800m hurdles is +0.41. d. If we were to measure Olympic gold medal times for the 800m hurdles in minutes instead of seconds, the correlation would be -0.66/60 = -0.011.

Maple trees A forester would like to know how big a maple tree might be at age 50 years. She gathers data from some trees that have been cut down, and plots the diameters (in inches) of the trees against their ages (in years). First she makes a linear model. The scatterplot and residuals plot are shown. a. Describe the association shown in the scatterplot. b. Do you think the linear model is appropriate? Explain. c. If she uses this model to try to predict the diameter of a 50-year old maple tree, would you expect that estimate to be fairly accurate, too low, or too high? Explain. Now she re-expresses the data, using the logarithm of age to try to predict the diameter of the tree. Here are the regression analysis and the residuals plot. Dependent variable is: Diam R squared =84.3\% Variable Coefficient s.e. of Coeff Constant -8.60770 1.681 Log(Age ) 15.0701 1.299 d. Explain why you think this is a better model. e. Using this model, predict the diameter of a maple tree at age 50 years.

The relationship between the longevity of an animal's life and its gestation time is 0.70. If an animal is one standard deviation below average in life expectancy, the gestation time is predicted to _____below average.

The residuals plot for a linear model is shown. Which is true?

All but one of the statements below contain a mistake. Which one could be true?

Halloween is a fun night. It seems that older children might get more candy because they can travel further while trick-or-treating. But perhaps the youngest kids get extra candy because they are so cute. Here are some data that examine this question, along with the regression output. Dependent Variable: candy Sample size: 9 $R$ (correlation coefficient) $=0.19534425$ $R-s q=0.038159375$ $s=11.297554$ Parameter Estimate Std. Err. Intercept 29.767327 12.21138 Age 0.88861386 1.6862229 -Based on the graph and the regression output, what conclusions do you draw regarding the relationship between age and the number of pieces of candy a trick-or-treater collects?

A consumer group collected information on HDTVs. They created a linear model to estimate the cost of an HDTV (in $) based on the screen size (in inches). Which is the most likely value of the slope of the line of best fit?

The following scatterplot shows the relationship between the time (in seconds) it took men to run the 1500m race for the gold medal and the year of the Olympics that the race was run in: a. Write a few sentences describing the association. b. Estimate the correlation. r =

If a data point is influential it……