Question
In: Statistics and Probability
Reported below are the amounts spent on advertising ($ millions) by a large firm from 2007...
Reported below are the amounts spent on advertising ($ millions) by a large firm from 2007 to 2017. Year Amount 2007 94.2 2008 96.9 2009 108.2 2010 109.3 2011 118.3 2012 129.5 2013 141.0 2014 141.5 2015 152.1 2016 163.1 2017 165.7 Click here for the Excel Data File Determine the logarithmic trend equation. (Do not round your intermediate calculations. Round your answers to 4 decimal places.) Estimate the advertising expenses for 2022. (Do not round intermediate calculations. Round your answer to 2 decimal places.) By what percent per year did advertising expense increase during the period? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
Solutions
Expert Solution
A.
| Amount( in $Mn) | Year(X) | Ln X | SUMMARY OUTPUT | |||||||||||
| 94.2 | 2007 | 7.6044 | ||||||||||||
| 96.9 | 2008 | 7.6049 | Regression Statistics | |||||||||||
| 108.2 | 2009 | 7.6054 | Multiple R | 0.993127536 | ||||||||||
| 109.3 | 2010 | 7.6059 | R Square | 0.986302303 | ||||||||||
| 118.3 | 2011 | 7.6064 | Adjusted R Square | 0.984780337 | ||||||||||
| 129.5 | 2012 | 7.6069 | Standard Error | 3.15021625 | ||||||||||
| 141.0 | 2013 | 7.6074 | Observations | 11 | ||||||||||
| 141.5 | 2014 | 7.6079 | ||||||||||||
| 152.1 | 2015 | 7.6084 | ANOVA | |||||||||||
| 163.1 | 2016 | 7.6089 | df | SS | MS | F | Significance F | |||||||
| 165.7 | 2017 | 7.6094 | Regression | 1 | 6431.107056 | 6431.107056 | 648.0447615 | 1.07187E-09 | ||||||
| Residual | 9 | 89.3147618 | 9.923862422 | |||||||||||
| Total | 10 | 6520.421818 | ||||||||||||
|
The Logarithm Trend equation of the line is Amount (Y) =Parameter estimate(b) * LnX + intercept(c) Amount (Y) = 15384.1591 * Ln X + -116896.4301 Amount (Y) = 15384.1591 * Ln X - 116896.4301 |
||||||||||||||
| Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | |||||||
| Intercept | -116896.4301 | 4597.037249 | -25.42864539 | 1.08243E-09 | -127295.6508 | -106497.2093 | -127295.6508 | -106497.2093 | ||||||
| Ln X | 15384.15909 | 604.3259732 | 25.45672331 | 1.07187E-09 | 14017.07876 | 16751.23942 | 14017.07876 | 16751.23942 | ||||||
| RESIDUAL OUTPUT | ||||||||||||||
| Amount( in $Mn) | Predicted Amount( in $Mn) | Residuals | ||||||||||||
| 94.2 | 90.81313458 | 3.386865424 | ||||||||||||
| 96.9 | 98.47647674 | -1.576476745 | ||||||||||||
| 108.2 | 106.1360035 | 2.063996542 | ||||||||||||
| 109.3 | 113.7917185 | -4.491718514 | ||||||||||||
| 118.3 | 121.4436257 | -3.143625703 | ||||||||||||
| 129.5 | 129.0917288 | 0.408271189 | ||||||||||||
| 141 | 136.7360316 | 4.26396838 | ||||||||||||
| 141.5 | 144.3765379 | -2.876537904 | ||||||||||||
| 152.1 | 152.0132514 | 0.086748568 | ||||||||||||
| 163.1 | 159.646176 | 3.453824032 | ||||||||||||
| 165.7 | 167.2753153 | -1.575315269 | ||||||||||||
The parameter estimate and the intercept can also be calculated by using the formulas in the excel.
| Calculated value | Formula Used | |
| c | 15384.15909 | c: =INDEX(LINEST(y,LN(x)),1) |
| b | -116896.4301 | b: =INDEX(LINEST(y,LN(x)),1,2) |
B. For 2022, the estimated advertising expenses is given by,
Amount (Y) = 15384.1591 * Ln X - 116896.4301
Amount (Y) = 15384.1591 * Ln 2022 - 116896.4301
Amount (Y) = 15384.1591 * 7.6118 - 116896.4301 = 205.3644
Therefore , Advertising expenses for 2022 is $ 205.36 Mn.
C. Advertising expenses in 2007 = 94.2
Advertising expenses in 2017 = 165.7
which means Advertising expenses have grown to 175.90% (= 165.7 / 94.2 ) of 2007 expenses in 2017
CAGR growth of Advertising expenses from 2007 to 2017 = (175.90%(1/10) - 1 ) = 105.81% - 1 = 5.81%
Therefore, Advertising expenses have grown by 5.81% from 2007 to 2017.
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