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September 30th, 2018, 08:22 AM   #1
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Exclamation Fitting Linear and Quadratic Functions to Data

Hello friends! I was hoping that you might be able to help me with these two questions that I have. I know it's a lot, so if you just want to help with a portion of one of the problems that is fine. All related comments are welcome. Thanks!

1. To determine the effect of alcohol consumption, a researcher has measured reaction times for
seven volunteers who were given different amounts of alcohol. The data is given below.
x = blood alcohol content
(percent)
y = reaction time
(seconds)
X Y
0.08 0.32
0.10 0.38
0.12 0.44
0.14 0.42
0.15 0.47
0.16 0.51
0.18 0.63

a) Make a scatter plot and comment on the strength of the linear relationship between the variables.

b) Find the linear function that best fits this data. Graph the line on the same axes with your scatter plot. Is it a good fit?

c) Use the model to predict the reaction times for an individual (not one of the above) whose blood alcohol content is 0.15 and 0.25.

d) Do you think these predictions are accurate? Explain


2. Proper inflation is necessary to maximize tread life for car tires. Consider the following values for pressure and tire life for a certain model of tire.

Pressure Tire life
(lb/in^2) (1000s of miles)

26 50
28 66
31 78
35 81
38 74
42 70
45 59

a) Make a scatter plot of the data. Is there a linear relationship between pressure and tire life or would a quadratic model be a better fit?

b) Find the quadratic function that best fits this data. Graph the function with your scatter plot.

c) Predict the mileage for a tire inflated to 30 psi.

d) Use the quadratic function to determine the optimum tire pressure to maximize tread life.
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October 1st, 2018, 06:59 PM   #2
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Math Focus: Yet to find out.
One question at a time and provide your working or state clearly what you are having trouble with. This will improve you chances of receiving help and you'll probably learn something too!
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October 1st, 2018, 09:41 PM   #3
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1.
(a) On making the scatter plot, the points seem to lie approximately on two lines. The first three points lie exactly on the line Y = 3X + 0.08. The remaining points lie approximately on the line Y = 5.24X - 0.318. Both of these are questionable models. There's too little data for my liking.

(b) The phrase "best fits" is not defined. What definition are you supposed to use, and why?

(c) As I've ducked part(b), I'll use my equation Y = 5.24X - 0.318 to predict the points (0.15, 0.47) and (0.25, 0.99) (rounded to 2 decimal places). The "official" answers are probably nearer (0.15, 0.49) and (0.25, 0.77).

(d) The given data is probably rather inaccurate and therefore not very suitable for making predictions, especially outside the range of the data.

2.
(a) There certainly doesn't appear to be a linear relationship. A quadratic model would be somewhat better, but a cubic model would be a further improvement, and hence indicates that there's little reason to prefer a quadratic model (other than that a linear model looks very inappropriate).

(b) Again, the term "best fit" needs to be defined.

(c) They're probably expecting a prediction of 70, but I prefer 70.7. It isn't stated what model is to be used for the prediction.

(d) 36 (lb/in²)
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