Reading formulas: a) We know, with a large sample size, it it easier to estmate
ID: 3200987 • Letter: R
Question
Reading formulas:
a) We know, with a large sample size, it it easier to estmate the mean of a population than an individual. So given the regression line y(hat) = B(hat)o + B(hat)1x, the CI for the average new value of y(hat) given a particular value of x called xp is given by Formula 1 and Formula 2.
b) We know, with a large sample size, it is easier to estimate the mean of a population than an individual. So given the regression line y(hat) =Bo + B1x, the confidence interval for an individual new value of y(hat) given a particular value of x called sp is given by Formula 1 Formula 2
CI Formula 1
with (n-2) degrees of freedom.
CI Formula 2
with (n-2) degrees of freedom.
Bonus: show the difference described above quite nicely
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Explanation / Answer
Answer:
Reading formulas:
Formula 2.
b) We know, with a large sample size, it is easier to estimate the mean of a population than an individual. So given the regression line y(hat) =Bo + B1x, the confidence interval for an individual new value of y(hat) given a particular value of x called sp is given by Formula 1
CI Formula 1
A 100(1-)% CI for an Individual New Value of y at x = xp:
y(hat) +/- (t/2)(s)1 + (1/n) + [((x(bar) - xp)2)/SSxx]
with (n-2) degrees of freedom.
CI Formula 2
A 100(1-)% CI for the mean value of y at x = xp:
y(hat) +/- (t/2)(s)(1/n) + [((xbar - xp0)2)/SSxx]
with (n-2) degrees of freedom.
A confidence interval is an interval associated with a parameter. A prediction interval is an interval associated with a random variable yet to be observed, with a specified probability of the random variable lying within the interval.
The difference between a prediction interval and a confidence interval is the standard error.
The standard error for a confidence interval on the mean takes into account the uncertainty due to sampling. The line we computed from our sample will be different from the line that would have been computed if we had the entire population, the standard error takes this uncertainty into account.
The standard error for a prediction interval on an individual observation takes into account the uncertainty due to sampling like above, but also takes into account the variability of the individuals around the predicted mean. The standard error for the prediction interval will be wider than for the confidence interval and hence the prediction interval will be wider than the confidence interval.
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