The number of initial public offerings of stock issued in a 10 year period and t
ID: 3222314 • Letter: T
Question
The number of initial public offerings of stock issued in a 10 year period and the total proceeds of these offerings (in millions) are shown in the table. Construct and interpret a 95% prediction interval for the proceeds when the number of issues is 614. The equation of the regression line is y^cap = 31.267x + 18.214.067. Construct and interpret a 95% prediction interval for the proceeds when the number of issues is 614. Select the correct choice below and fill in the answer boxes to complete your choice (Round to the nearest dollars as needed.) There is a 95% chance that the predicted proceeds given 614 issues is between $ and $ We can be 95% confident that when there are 811 issues, the proceeds will be between $ and $ .Explanation / Answer
there are n=10 values of x and y
and the regression equation is given as
y=31.267x+18214.067
we want to find a 95% confidence interval for predicted y when x=614
the anova table is given as
Source DF SS MS F P
Regression 1 392529871 392529871 8.07 0.022
Residual Error 8 389167471 48645934
Total 9 781697342
so MSE=48645934
now xbar=mean of x values=340.3
and sxx=sum(x-xbar)2=401508
as we want to find 95% confidence interval for n=10
so the upper 0.025 point of a t distribution with df=n-2=8 be t0.025;8= 2.306
when x=614 the value of y from regression equation is
yhat=31.267*614+18214.067=37412.005
so 95% confidence interval for predicted y when x=614 is
[yhat-t0.025;8*sqrt(MSE*(1/n+(614-xbar)2/sxx)),yhat+t0.025;8*sqrt(MSE*(1/n+(614-xbar)2/sxx))]
=[37412.005-2.306*sqrt(48645934*(1/10+(614-340.3)2/401508)),37412.005+2.306*sqrt(48645934*(1/10+(614-340.3)2/401508))]
=[37412.005-8609.984,37412.005+8609.98]=[28802.021,46021.985] (in millions)
=[28802.021*1000000,46021.985*1000000]=[28802021000,46021985000] [answer]
and since the random quantity is the confidence interval and the predicted proceeds here is the parameter
so the chance is associated with the confidence interval
hence correct answer is
B. we can be 95% confident that when there are 614 issues, the proceeds will be between $28802021000 and $46021985000 [answer]