Consider the simple linear regression model yi=β0+β1xi+ϵi, where ϵi's are independent n(0,σ2) random variables. therefore, yi is a normal random variable with mean β0+β1xi and variance σ2. moreover, yi's are independent. as usual, we have the observed data pairs (x1,y1), (x2,y2), ⋯⋯, (xn, yn) from which we would like to estimate β0 and β1. in this chapter, we found the following estimators β1^=sxysxx,β0^=y−β1^x¯¯¯. where sxx=∑i=1n(xi−x¯¯¯)2,sxy=∑i=1n(xi−x¯ ¯¯)(yi−y). show that β1^ is a normal random variable. show that β1^ is an unbiased estimator of β1, i. e., e[β1^]=β1. show that var(β1^)=σ2sxx.

See proof below.

Step-by-step explanation:

If we assume the following linear model:

And if we have n sets of paired observations the model can be written like this:

And using the least squares procedure gives to us the following least squares estimates for and for  :

Where:

Then is a random variable and the estimated value is . We can express this estimator like this:

Where and if we see careful we notice that and

So then when we find the expected value we got:

And as we can see is an unbiased estimator for

In order to find the variance for the estimator we have this:

And we can assume that since the observations are assumed independent, then we have this:

And if we simplify we got:

And with this we complete the proof required.

answer: city needs 2.4 inches of rainfall more to match the record of 40 inches.

step-by-step explanation:

since we have given that

total annual rainfall a city record = 40 inches

this year, total rainfall recorded = 37.6 inches

number of inches of rainfall   is needed to match the record of 40 inches is given by

hence, city needs 2.4 inches of rainfall more to match the record of 40 inches.

thats one

step-by-step explanation: