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 serialcorrelation 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 aboutserial correlation, see the book Ibbotson SBBI published by Morningstar.An Internet advertising agency is studying the number of "hits" on a certain web site during an advertising campaign. It is hoped that as the campaign progresses, the number of hits on the web sitewill also increase in a predictable way from one day to the next. For 10 days of the campaign, the number of hits x 10° is shown.Original Time SeriesDay12478.9.10Hits x 1051.33.54.47.26.98.29.011.213.114.7(a) To construct a serial correlation, we use data pairs (x, y) where x =original data and yoriginal data shifted ahead by one time period. Construct the data set (x, y) for serial correlationby filling in the following table.1.33.54.47.26.98.29.011.213.1y(b) For the (x, y) data set of part (a), compute the equation of the sample least-squares line ý= a + bx. (Use 4 decimal places.)abIf the number of hits was 9.3 (× 10°) one day, what do you predict for the number of hits the next day? (Use 1 decimal place.)(x 10°) hits(c) Compute the sample correlation coefficient r and the coefficient of determination r. (Use 4 decimal places.)Test p > 0 at the 1% level of significance. (Use 2 decimal places.)critical t

(a) Let x be the original data, y be the original data shifted ahead by one time period. The data table is, x 1.3 3.5 4.4 7.2 6.9 8.2 9.0 11.2 13.1 y 3.5 4.4 7.2 6.9 8.2 9.0 11.2 13.1 14.7 (b) The regression analysis is conducted using EXCEL. The software procedure is given below: Enter the data. Select Data > Data Analysis >Regression> OK. Enter Input Y Range as y. Enter Input X Range as x. Mark Labels. Click OK. The output using EXCEL is as follows: From the output, the intercept a is 1.6659, and slope b is 0.9754. The equation of the sample least-squares line is, y^=a+bx=1.6659+0.9754x Thus, the equation of the sample least-squares line is y^=1.6659+0.9754x. The predicted value for the number of hits the next day when the number of hits was 9.3 one day is, y^=1.6659+0.97549.3=1.6659+9.07122=10.73712≈10.7 Thus, the predicted value for the number of hits the next day when the number of hits was 9.3 one day is 10.7.(c) The correlation value is obtained using EXCEL. The software procedure is given below: Enter the data. Select Data > Data Analysis >Correlation> OK. Enter Input Range as A1:B10. Mark Labels in First Row. Click OK. The output using EXCEL is as follows: From the output, the sample correlation coefficient is 0.9690. Thus, the sample correlation coefficient is 0.9690. The coefficient of determination is, r2=0.96902=0.938961≈0.9390 Thus, the coefficient of determination is 0.9390. The hypothesis is, Null hypothesis: H0:ρ≤0 Alternative hypothesis: H1:ρ>0 The test statistic is, t=rn-21-r2=0.96909-21-0.96902=10.38 Thus, the test statistic t is 10.38. The degrees of freedom is, df=n-2=9-2=7 The degrees of freedom is 7. Computation of critical value: The critical value of t-distribution for 0.99 area at 7 degrees of freedom can be obtained using the excel formula “=T.INV(0.99,7)”. The critical value is 3.00. Thus, the critical value is 3.00. Decision rule: If test statistic > critical value, then reject the null hypothesis. Otherwise, do not reject the null hypothesis. Conclusion: The test statistic (10.38) is greater than the critical value (3.00). Based on decision rule, reject the null hypothesis. Thus, there is sufficient evidence that the ρ>0.

Math

Statistics

(a) To construct a serial correlation, we use data pairs (x, y) where x = original data and y = original data shifted ahead by one time period. Construct the data set (x, y) for serial correlation by filling in the following table. 1.3 3.5 4.4 7.2 6.9 8.2 9.0 11.2 13.1 y (b) For the (x, y) data set of part (a), compute the equation of the sample least-squares line ŷ = a + bx. (Use 4 decimal places.) a b If the number of hits was 9.3 (x 105) one day, what do you predict for the number of hits the next day? (Use 1 decimal place.) |(x 105) hits (c) Compute the sample correlation coefficient r and the coefficient of determination r2. (Use 4 decimal places.) Test p > 0 at the 1% level of significance. (Use 2 decimal places.) critical t

(a) To construct a serial correlation, we use data pairs (x, y) where x = original data and y = original data shifted ahead by one time period. Construct the data set (x, y) for serial correlation by filling in the following table. 1.3 3.5 4.4 7.2 6.9 8.2 9.0 11.2 13.1 y (b) For the (x, y) data set of part (a), compute the equation of the sample least-squares line ŷ = a + bx. (Use 4 decimal places.) a b If the number of hits was 9.3 (x 105) one day, what do you predict for the number of hits the next day? (Use 1 decimal place.) |(x 105) hits (c) Compute the sample correlation coefficient r and the coefficient of determination r2. (Use 4 decimal places.) Test p > 0 at the 1% level of significance. (Use 2 decimal places.) critical t

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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