erial correlation, also known asautocorrelation, describes the extent to which the result in one period of a time series is related to the result in the next period. A time series with high serial correlation is said to be very predictable from one period to the next. If the serial correlation is low (or near zero), the time series is considered to be much less predictable. For more information about serial correlation, see the bookIbbotson SBBIpublished by Morningstar.

A research veterinarian at a major university has developed a new vaccine to protect horses from West Nile virus. An important question is: How predictable is the buildup of antibodies in the horse's blood after the vaccination is given? A large random sample of horses were given the vaccination. The average antibody buildup factor (as determined from blood samples) was measured each week after the vaccination for 8 weeks. Results are shown in the following time series.

Original Time SeriesWeek12345678Buildup Factor2.14.76.27.58.09.310.712.4

To construct a serial correlation, we simply use data pairs(x,y)

wherex= originalbuildup factor data andy= originaldata shifted ahead by1 week.This gives us the following data set. Since we are shifting1 weekahead, we now have7 datapairs (not 8).

Data for Serial Correlationx2.14.76.27.58.09.310.7y4.76.27.58.09.310.712.4

For convenience, we are given the following sums.

x=48.5,

y=58.8,

x²=386.17,

y²=535.52,

xy=452.1

(a) Use the sums provided (or a calculator with least-squares regression) to compute the equation of the sample least-squares line,=a+bx.

(Use 4 decimal places.)a

b

If the buildup factor wasx=5.9

one week, what would you predict the buildup factor to be the next week? (Use 2 decimal places.)

(b) Compute the sample correlation coefficientrand the coefficient of determinationr².

(Use 4 decimal places.)r

r²

Test> 0

at the1% levelof significance. (Use 2 decimal places.)tcriticaltConclusion

Reject the null hypothesis, there is sufficient evidence that> 0.

Reject the null hypothesis, there is insufficient evidence that> 0.

Fail to reject the null hypothesis, there is insufficient evidence that> 0.

Fail to reject the null hypothesis, there is sufficient evidence that> 0.

Would you say the time series of antibody buildup factor is relatively predictable from one week to the next? Explain.

Yes, the data support a high positive serial correlation and indicate a predictable original time series from one week to the next.

No, the data do not support a high serial correlation and do not indicate a predictable original time series from one week to the next.

Yes, the data support a high negative serial correlation and indicate a predictable original time series from one week to the next.