Statistics, prediction

[MaxManus]

Homework Statement

Given a simple linear model y = B₁ + B₂*x + e and the least square estimators, we can estimate E(y) for any value of x = x₀ as Y₀ = b₁ + b₂*x₀

Describe the difference between predicting y₀ and estimating E(y₀)

Homework Equations

The Attempt at a Solution

I am not sure what the difference is.

Prediction:
Y= b₁ + b₂*x₀

f = y₀ - Y₀ = (B₁ +B₂*x₀ + e₀) - (b₁ + b₂*x₀)

E(f) = 0

and var(f) = sigma²*(1 + 1/N + (x₀ - $$\bar{x}$$)²/sum(x_i - $$\bar{}x$$)²

where $$\bar{}x$$ is the average.

the prediction interval is Y₀ +- t_c*se(f)

where se is the standard error of the forcast.

Can anyone help me with estimating E(y₀)?

 

[mighty2000]

Estimating E(y0) is different from predicting y0 in that estimating E(y0) involves using statistical methods to determine the expected value or mean of the response variable y at a specific value of the predictor variable x. This is typically done using a regression model, such as the simple linear model provided in the forum post. In this case, the estimated value of E(y0) would be the point estimate of the mean response at x0, denoted as Y0.

On the other hand, predicting y0 involves using the regression model to make a specific prediction for the response at a given value of the predictor variable. In this case, the predicted value of y0 would be denoted as Y0 and would take into account not only the estimated mean response, but also the random error term e. This allows for a range of possible values for y0, rather than a single point estimate.

In summary, estimating E(y0) involves determining the expected value of the response variable at a specific value of the predictor variable, while predicting y0 involves making a specific prediction for the response at that value of the predictor variable, taking into account the inherent uncertainty in the model.