Question

Which of the following statements is true about prediction intervals and confidence intervals for Y in simple linear regression?

A) A 95% prediction interval for Y given a specific value of the X variable in a regression model is calculated using the formula

y with bar on top plus-or-minus 2 fraction numerator s over denominator square root of n end fraction,where s is the sample standard deviation,y with bar on topis the sample mean, and n is the sample size. A confidence interval for the mean value of Y replaces the value of "2" with 1.96.

B) If a 95% prediction interval contains 0, we will be able to reject the hypothesisH subscript 0 : beta subscript 1 equals 0at the 0.05 level of significance.

C) A prediction interval gives a range of "reasonable" values for the actual (new) value of the Y variable, given a specific choice of the X variable. It will, in general, always be narrower than a confidence interval for the mean value of the Y variable because it is easier in general to predict a specific value than to estimate a mean.

D) A prediction interval gives a range of "reasonable" values for an actual (new) value of the Y variable, given a specific choice of the X variable, while a confidence interval gives a range of "reasonable values" for the mean of the Y variable, given a specific choice of the X variable.

E) A prediction interval gives a range of "reasonable" values for the mean value of the Y variable, given a specific choice of the X variable, while a confidence interval gives a range of "reasonable values" for an actual (new) value of the Y variable, given a specific choice of the X variable.

Answer 1

The true statement about prediction intervals and confidence intervals in simple linear regression is that a prediction interval provides a range for a new actual value of Y based on X, while a confidence interval provides a range for the mean value of Y based on X.

Step-by-step explanation:

The statement that is true about prediction intervals and confidence intervals for Y in simple linear regression is:

D) A prediction interval gives a range of "reasonable" values for an actual (new) value of the Y variable, given a specific choice of the X variable, while a confidence interval gives a range of "reasonable values" for the mean of the Y variable, given a specific choice of the X variable.

Confidence intervals and prediction intervals are both centered around the predicted value of Y given X, but they have different purposes. A confidence interval is constructed around the mean of the Y values for a given X and reflects the uncertainty around the estimation of this mean. The interval is calculated based on the assumption that if we took many samples, a certain percentage (for example, 95%) of those confidence intervals calculated from those samples would contain the true population mean.

In contrast, a prediction interval is used to predict where an individual new observation might fall, given a specific value of X. This interval is wider than the confidence interval because it also accounts for the variability of individual observations around the predicted mean.