In the casting of metal parts, molten metal flows through a “gate” into a die that shapes the part. The gate velocity (the speed at which metal is forced through the gate) plays a critical role in die casting. A firm that casts cylindrical aluminum pistons examined a random sample of pistons formed from the same alloy of metal. What is the relationship between the cylinder wall thickness (inches) and the gate velocity (feet per second) chosen by the skilled workers who do the casting? If there is a clear pattern, it can be used to direct new workers or automate the process. A scatterplot of the data is shown below.
A least-squares regression analysis was performed on the data. Some computer output and a residual plot are shown below. A Normal probability plot of the residuals (not shown) is roughly linear.
(a) Describe what the scatterplot tells you about the relationship between cylinder wall thickness and gate velocity.
(b) What is the equation of the least-squares regression line? Define any variables you use.
(c) One of the cylinders in the sample had a wall thickness of inches. The gate velocity chosen for this cylinder was feet per second. Does the regression line in part (b) overpredict or underpredict the gate velocity for this cylinder? By how much? Show your work.
(d) Is a linear model appropriate in this setting? Justify your answer with appropriate evidence.
(e) Interpret each of the following in context:
(i) The slope
(iv) The standard error of the slope
(a) The scatter plot diagram is Positive, Linear, and fairly Solid.
(b) The regression line equation is and the variables are the thickness, the velocity.
(c) The expected value of is thus greater than , indicating that the gate velocity has been overestimated.
(d) Yes, there is a linear model appropriate in this setting.
(e) The values of
The plot slopes upward, hence the direction is positive.
Because the points sit uniformly along a line and there is no curvature, the form is linear.
Strength: Good, because the points aren't too far apart or too near together.
The coefficient of and are:
General regression equation
From the part (b)
Replacing the value of
Yes, because the residual plot shows no discernible trend and the residual in the plot is centered at roughly on the basis of the presented figure.
(i) The slope is estimated to be , therefore a gate speed raise of per inch of thickness is necessary.
(ii) In other words, assumptions are predicted to differ from the genuine value on average.
(iii) This means the least-square regression line explained of the variance between the variables.
(iv) This means that the population regression line's anticipated slope would deviate by an average of .
A residual plot from the least-squares regression is shown below. Which of the following statements is supported by the graph
(a) The residual plot contains dramatic evidence that the standard deviation of the response about the population regression line increases as the average number of putts per round increases.
(b) The sum of the residuals is not 0. Obviously, there is a major error present.
(c) Using the regression line to predict a player’s total winnings from his average number of putts almost always results in errors of less than .
(d) For two players, the regression line under predicts their total winnings by more than.
(e) The residual plot reveals a strong positive correlation between average putts per round and prediction errors from the least-squares line for these players.
Western lowland gorillas, whose main habitat is the central African continent, have a mean weight of with a standard deviation of . Capuchin monkeys, whose main habitat is Brazil and a few other parts of Latin America, have a mean weight of with a standard deviation of . Both weight distributions are approximately Normally distributed. If a particular western lowland gorilla is known to weigh , approximately how much would a capuchin monkey have to weigh, in pounds, to have the same standardized weight as the lowland gorilla?
(e) There is not enough information to determine the weight of a capuchin monkey.
The slope of the population regression line describes
(a) the exact increase in the selling price of an individual unit when its appraised value increases by
(b) the average increase in the appraised value in a population of units when the selling price increases by $51000.
(c) the average increase in selling price in a population of units when the appraised value increases by $ 1000.
(d) the average selling price in a population of units when a unit's appraised value is 0.
(e) the average increase in appraised value in a sample of 16 units when the selling price increases by $ 1000.
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