# Normal Probability for a Linear Regression

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• Dec 5th 2012, 03:58 PM
xxlvh
Normal Probability for a Linear Regression
Suppose that the purity of a chemical solution y is related to the amount of a catalyst x through a linear regression model y = 123 -2.16x with a standard deviation of 4.1. What is the probability that the purity of a solution with a catalyst level of 30 is smaller than the purity of a catalyst level of 27.5?
• Dec 5th 2012, 04:27 PM
chiro
Re: Normal Probability for a Linear Regression
Hey xxlvh.

The first thing you have to think about is the distribution and how to obtain the probability.

If you are given a standard error, then consider that the distribution of (y - E[y])/SE(y) ~ N(0,1) or standard normal.

Now the next thing becomes translating your statement into a probability: what do you think this is with the information given to you?
• Dec 5th 2012, 04:31 PM
xxlvh
Re: Normal Probability for a Linear Regression
I don't understand?

I envisioned the problem as two different bell curves, one with the expected mean for x=30 and another with the expected mean for x=27.5
• Dec 5th 2012, 04:35 PM
chiro
Re: Normal Probability for a Linear Regression
Perhaps you could clarify the probability statement you are thinking of in terms of the random variables X and Y.
• Dec 5th 2012, 04:38 PM
xxlvh
Re: Normal Probability for a Linear Regression
P(n(123+(-2.16*30),4.1^2) < n(123+(-2.16*27.5),4.1^2))
• Dec 5th 2012, 04:41 PM
chiro
Re: Normal Probability for a Linear Regression
What is the quantity n? Is it a random variable? If so what distribution does it take?
• Dec 5th 2012, 05:49 PM
xxlvh
Re: Normal Probability for a Linear Regression
Pardon the lack of LaTex, I forgot how to use it..I mean't N, as in normal probability...
• Dec 5th 2012, 06:07 PM
chiro
Re: Normal Probability for a Linear Regression
Is the N just a standard normal N ~ N(0,1)?
• Dec 5th 2012, 06:23 PM
xxlvh
Re: Normal Probability for a Linear Regression
No, I stated before the properties. One has mean of the expected value at 30, one has a mean of the expected value of 27.5 and both have standard deviations of 4.1
• Dec 5th 2012, 06:25 PM
xxlvh
Re: Normal Probability for a Linear Regression
I am near positive I am interpreting the question right but I don't know how to begin solving for the probability.
• Dec 5th 2012, 07:17 PM
chiro
Re: Normal Probability for a Linear Regression
If you want to solve for the probability you need a distribution.

Parameters in a regression usually have a t-distribution with the appropriate degrees of freedom.
• Dec 5th 2012, 07:19 PM
xxlvh
Re: Normal Probability for a Linear Regression
No, you wouldn't use a t-distribution if the standard deviation is given.
• Dec 5th 2012, 07:19 PM
xxlvh
Re: Normal Probability for a Linear Regression
Which is why I used the Normal Distribution. It's from a textbook section on Normal Probabilities
• Dec 5th 2012, 08:31 PM
chiro
Re: Normal Probability for a Linear Regression