Stats Online Ch. 7
The correlation between a cereal's fiber and potassium contents is r=0.876. What fraction of the variability in potassium is accounted for by the amount of fiber that servings contain?
0.876^2 = 0.7673*100= 76.7%
A student in an intro stats course collects data at her university. She wants to model the relationship between student jobs and GPA. She collects a random sample of students and asks each for their GPA and the number of hours per week they work. She checks the conditions and makes a linear model with GPA as the response variable. She finds that the R-squared statistic is 12.7%. What is the correct interpretation of this number?
12.7% of changes in GPA can be explained by differences in student work hours.
For many people, breakfast cereal is an important source of fiber in their diets. Cereals also contain potassium, a mineral shown to be associated with maintaining a healthy blood pressure. An analysis of the amount of fiber (in grams) and the potassium content (in milligrams) in servings of 77 breakfast cereals produced the regression model Potassium with caret equals 37 plus 26 Fiber. If your cereal provides 9 grams of fiber per serving, how much potassium does the model estimate you will get?
37+26(9) = 271
A random sample of records of sales of homes in a city gives the Price and Size (in square feet) of 117 homes. A regression to predict Price (in thousands of dollars) from Size has an R squared of 71.4%. Write a sentence in context summarizing what the R squared says about this regression.
A linear model on Size accounts for 71.4% of the variation in home Price.
An analysis of the amount of fiber (in grams) and the potassium content (in milligrams) in servings of 77 breakfast cereals produced the regression model Potassium with caret=36+ 27 Fiber. Explain what the slope means.
The model predicts that cereals will have approximately 27 more milligrams of potassium for every additional gram of fiber.
For many people, breakfast cereal is an important source of fiber in their diets. Cereals also contain potassium, a mineral shown to be associated with maintaining a healthy blood pressure. An analysis of the amount of fiber (in grams) and the potassium content (in milligrams) in serving of 77 breakfast cereals produced the regression model Potassium with caret =38 + 27 Fiber . From this model you can estimate a cereal's potassium content from the amount of fiber it contains. In this context, what does it mean to say that a cereal has a negative residual?
The potassium content is actually lower than the model predicts for a cereal with that much fiber.
If the correlation between two variables is positive, the slope of the linear model will be positive.
True
Standardized test scores are often used as part of an application to college. Test scores in math and verbal are between 200 and 800 but have no units.
a)Verbal and math test scores of a high school graduating class are displayed in the scatterplot, with the regression line added. Describe the relationship. The association between math scores and verbal scores is moderately linear and positive. Students with high verbal scores typically have high math scores. b)Are there any students whose scores do not seem to fit the overall pattern? Yes, one student got a verbal score of 250 and a math score of 800. c)For these data, r=0.623. Interpret this statistic. Students who score one standard deviation above the mean in verbal are expected to score 0.623 standard deviations above the mean in math. Also, 38.8% of the variability in math score is explained by variability in verbal score. d)These verbal scores averaged 470.8, with a standard deviation of 175.7, and the math scores averaged 465.4, with a standard deviation of 172.9. Write the equation of the regression line. .623*172.9/175.7=0.613 465.4-(0.613*470.8)=176.800 y^ = 176.800 + 0.613 x e)Interpret the slope of this line. Recall that y^ = bo + b1x. For each additional point in verbal, the model predicts an increase of b 1 points in math score. f)Predict the math score of a student with a verbal score of 509. The student is expected to have a math score of 176.800 + 0.613(509) = 488 g)Every year some student scored a perfect 1600 (combined math and verbal score). Based on this model, what would that student's residual be for her math score? 176.800 + 0.613(800) = 1157.60
The manager of a bookstore wants to predict Sales from Number of Sales People Working using the accompanying data set. What is the value of R squared and what does it mean?
r= 0.973 = .973^2 = .9467=94.7% The value of Upper R squared is """94.7%""", which is the percentage of variance in Sales that can be accounted for by the regression of Sales on Number of Sales People Working.
The data below show the Fat and Calories in fast-food hamburgers. A model was created that can estimate the number of Calories in a burger when the Fat content is known. Fat (g) Calories 19 410 32 585 33 595 35 570 39 645 40 675 43 665
a) Explain why you cannot use that model to estimate the fat content of a burger with 600 calories. The regression model was for predicting calories from fat, not the other way around. b) Using an appropriate model, estimate the fat content of a burger with 600 calories. -15.900+0.085*x -15.900+0.085(600)=35.1
A random sample of records of home sales from Feb. 15 to Apr. 30, 1993, from the files maintained by the Albuquerque Board of Realtors gives the Price and Size (in square feet) of 117 homes. A regression to predict Price (in thousands of dollars) from Size has an Upper R squared of 71.4 %. The residuals plot indicated that a linear model is appropriate.
a) What are the variables and units in this regression? Price (in thousands of dollars) is y and Size (in square feet) is x. b) What units does the slope have? The slope has units of thousands of dollars per square foot. c) Do you think the slope is positive or negative? The slope is positive. As the size of the home increases, the price should also increase.
A regression analysis of 117 homes for sale produced the following model, where price is in thousands of dollars and size is in square feet. Price with caret=47.83+0.068(Size)
a)Explain what the slope of the line says about housing prices and house size. For every additional square foot of area of a house, the price is predicted to increase by $68. b) What price would you predict for a 2500-square-foot house in this market? 47.83 + 0.068(2500) = 217.83 217.83*1000= $217,830 c) A real estate agent shows a potential buyer a 1300-square-foot house, saying that the asking price is $6000 less than what one would expect to pay for a house of this size. What is the asking price? 47.83 + 0.068(1300) = 136.23 * 1000 = 136,230 136,230 - 6000 = 130,230 $6000 per square foot What is the $5500 called? Residual
