Question

1)

(a) Suppose n = 6 and the sample correlation coefficient is r = 0.874. Is r significant at the 1% level of significance (based on a two-tailed test)? (Round your answers to three decimal places.)

t	=
critical t	=

Conclusion:

Yes, the correlation coefficient ρ is significantly different from 0 at the 0.01 level of significance.No, the correlation coefficient ρ is not significantly different from 0 at the 0.01 level of significance.    

(b) Suppose n = 10 and the sample correlation coefficient is r = 0.874. Is r significant at the 1% level of significance (based on a two-tailed test)? (Round your answers to three decimal places.)

t	=
critical t	=

Conclusion:

Yes, the correlation coefficient ρ is significantly different from 0 at the 0.01 level of significance.No, the correlation coefficient ρ is not significantly different from 0 at the 0.01 level of significance.    

(c) Explain why the test results of parts (a) and (b) are different even though the sample correlation coefficient r = 0.874 is the same in both parts. Does it appear that sample size plays an important role in determining the significance of a correlation coefficient? Explain.

As n decreases, the degrees of freedom and the test statistic increase. This produces a smaller P value.As n increases, so do the degrees of freedom, and the test statistic. This produces a smaller P value.    As n increases, so do the degrees of freedom, and the test statistic. This produces a larger P value.As n increases, the degrees of freedom and the test statistic decrease. This produces a smaller P value.

2)

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	9.1	10.6	12.4

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	9.1	10.6
y	4.6	6.2	7.5	8.0	9.1	10.6	12.4

For convenience, we are given the following sums.

Σx = 48.3,

Σy = 58.4,

Σx² = 380.31,

Σy² = 528.78,

Σxy = 446.3

(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.

(Use 4 decimal places.)

a
b

If the buildup factor was

x = 5.2

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


(b) Compute the sample correlation coefficient r and the coefficient of determination

r².

(Use 4 decimal places.)

r
r2

Test

ρ > 0

at the 1% level of significance. (Use 2 decimal places.)

t
critical t

Conclusion

Reject the null hypothesis, there is sufficient evidence that ρ > 0.Reject the null hypothesis, there is insufficient evidence that ρ > 0.    Fail to reject the null hypothesis, there is insufficient evidence that ρ > 0.Fail to reject the null hypothesis, there is sufficient evidence that ρ > 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 negative serial correlation and indicate a predictable original time series from one week to the next.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.

Answer 1

1)

a)

n=   6                          
alpha,α =    0.01                          
correlation , r =   0.8740                          
                              
Df =    n-2 =   4                      
t-test statistic = r*√(n-2)/√(1-r²) =        0.8740   * √    4   / √ ( 1 -    0.8740   ² ) =   3.597
                              

                              
critical t-value =    4.604   [excel function: =t.inv.2t(α,df) ]                       

No, the correlation coefficient ρ is not significantly different from 0 at the 0.01 level of significance.

b)

t-test statistic = r*√(n-2)/√(1-r²) =        0.8740   * √    8   / √ ( 1 -    0.8740   ² ) =   5.087
critical t-value =    3.355
Yes, the correlation coefficient ρ is significantly different from 0 at the 0.01 level of significance

c) As n increases, so do the degrees of freedom, and the test statistic. This produces a smaller P value.