Serial correlation, also known as autocorrelation, 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 book Ibbotson SBBI published 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 Series Week 1 2 3 4 5 6 7 8 Buildup Factor 2.3 4.6 6.2 7.5 8.0 9.4 10.6 12.1 To construct a serial correlation, we simply use data pairs (x, y) where x = original buildup factor data and y = original data shifted ahead by 1 week. This gives us the following data set. Since we are shifting 1 week ahead, we now have 7 data pairs (not 8). Data for Serial Correlation x 2.3 4.6 6.2 7.5 8.0 9.4 10.6 y 4.6 6.2 7.5 8.0 9.4 10.6 12.1 For convenience, we are given the following sums. Σx = 48.6, Σy = 58.4, Σx2 = 385.86, Σy2 = 526.98, Σxy = 448.7 (a) Use the sums provided (or a calculator with least-squares regression) to compute the equation of the sample least-squares line, ŷ = a + bx. (Round your answers to four decimal places.) a=? b= .8926 Compute the sample correlation coefficient r and the coefficient of determination r2. (Round your answers to four decimal places.) r=? r2=? Test ? > 0 at the 1% level of significance. (Round your answers to three decimal places.) t=? critical t=?

Serial correlation, also known as autocorrelation, 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 book Ibbotson SBBI published 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 Series Week 1 2 3 4 5 6 7 8 Buildup Factor 2.3 4.6 6.2 7.5 8.0 9.4 10.6 12.1 To construct a serial correlation, we simply use data pairs (x, y) where x = original buildup factor data and y = original data shifted ahead by 1 week. This gives us the following data set. Since we are shifting 1 week ahead, we now have 7 data pairs (not 8). Data for Serial Correlation x 2.3 4.6 6.2 7.5 8.0 9.4 10.6 y 4.6 6.2 7.5 8.0 9.4 10.6 12.1 For convenience, we are given the following sums. Σx = 48.6, Σy = 58.4, Σx2 = 385.86, Σy2 = 526.98, Σxy = 448.7 (a) Use the sums provided (or a calculator with least-squares regression) to compute the equation of the sample least-squares line, ŷ = a + bx. (Round your answers to four decimal places.) a=? b= .8926 Compute the sample correlation coefficient r and the coefficient of determination r2. (Round your answers to four decimal places.) r=? r2=? Test ? > 0 at the 1% level of significance. (Round your answers to three decimal places.) t=? critical t=?

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

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Chapter4: Equations Of Linear Functions

Section4.5: Correlation And Causation

Problem 2CYU

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Serial correlation, also known as autocorrelation, 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 book Ibbotson SBBI published 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 Series

Week	1	2	3	4	5	6	7	8
Buildup Factor	2.3	4.6	6.2	7.5	8.0	9.4	10.6	12.1

To construct a serial correlation, we simply use data pairs

(x, y)

where x = original buildup factor data and y = original data shifted ahead by 1 week. This gives us the following data set. Since we are shifting 1 week ahead, we now have 7 data pairs (not 8).

Data for Serial Correlation

x	2.3	4.6	6.2	7.5	8.0	9.4	10.6
y	4.6	6.2	7.5	8.0	9.4	10.6	12.1

For convenience, we are given the following sums.

Σx = 48.6,

Σy = 58.4,

Σx² = 385.86,

Σy² = 526.98,

Σxy = 448.7

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

ŷ = a + bx.

(Round your answers to four decimal places.)

a=?

b= .8926

Compute the sample correlation coefficient r and the coefficient of determination r².

(Round your answers to four decimal places.)

r=?

r²=?

Test ? > 0 at the 1% level of significance. (Round your answers to three decimal places.)

t=?

critical t=?

Expert Solution

Step 1

Given that;

$\sum_{} x = 48.6 \sum_{} y = 58.4 \sum_{} x^{2} = 385.86 \sum_{} y^{2} = 526.98 \sum_{} x y = 448.7$

(a) The slope b is given by;

$b = \frac{n (\sum_{} x y) - (\sum_{} x) (\sum_{} y)}{n (\sum_{} x^{2}) - {(\sum_{} x)}^{2}} = \frac{7 (448.7) - [(48.6) (58.4)]}{7 (385.86) - {(48.6)}^{2}} = \frac{302.66}{339.06} = 0.8926$

The y-intercept 'a' is given by;

$a = \bar{y} - b \bar{x} = \frac{\sum_{} y}{n} - (0.8926 \times \frac{\sum_{} x}{n}) = \frac{58.4}{7} - (0.8926 \times \frac{48.6}{7}) = 2.1457$

Hence the sample least square line is given by;

$\hat{y} = 2.1457 + 0.8926 x$

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