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?

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?

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

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.

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))

Re: Normal Probability for a Linear Regression

What is the quantity n? Is it a random variable? If so what distribution does it take?

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...

Re: Normal Probability for a Linear Regression

Is the N just a standard normal N ~ N(0,1)?

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

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.

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.

Re: Normal Probability for a Linear Regression

No, you wouldn't use a t-distribution if the standard deviation is given.

Re: Normal Probability for a Linear Regression

Which is why I used the Normal Distribution. It's from a textbook section on Normal Probabilities

Re: Normal Probability for a Linear Regression

Did you get your intended answer?

Re: Normal Probability for a Linear Regression

No, I don't know how to start solving the question