Complete the following table, showing the moving averages for the number of orders based on a five-observation cycle. The first one has been done for you. Input all moving averages to one decimal place.
\n\n| Mon | Tues | Thur | Fri | Sat | |
|---|---|---|---|---|---|
| 7/4/08 | \n* | \n* | \n$\\var{mth1}$ | \n[[0]] | \n[[1]] | \n
| 14/4/08 | \n[[2]] | \n[[3]] | \n[[4]] | \n[[5]] | \n[[6]] | \n
| 21/4/08 | \n[[7]] | \n[[8]] | \n[[9]] | \n* | \n* | \n
Fit the linear regression model $Y=\\alpha+\\beta T+\\epsilon$ to this set $Y$ of moving averages, where $T$ represents time. ($T=1$ for 7/4/08, $T=2$ for 8/4/08, $T=3$ for 10/4/08 etc.)
\nYou may use the following summaries:
\n$\\sum t=88,\\;\\;\\sum y=\\var{summa},\\;\\;\\sum ty=\\var{sumty},\\;\\; \\sum t^2=814,\\;\\;\\sum y^2=\\var{sumysquared}$.
\nEstimate for $\\beta= \\;$[[0]] (estimate to 2 decimal places).
\nEstimate for $\\alpha= \\;$[[1]] (estimate to 2 decimal places). Use the estimate for $\\beta$ to 2 decimal places to estimate $\\alpha$.
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "estbe-tol", "maxValue": "estbe+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "estal-tol", "maxValue": "estal+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "Complete the following table, showing the seasonal deviations and seasonal means for each day of the week:
\n| Mon | Tues | Thur | Fri | Sat | |
|---|---|---|---|---|---|
| 7/4/08 | \n* | \n* | \n[[0]] | \n[[1]] | \n[[2]] | \n
| 14/4/08 | \n[[3]] | \n[[4]] | \n[[5]] | \n[[6]] | \n[[7]] | \n
| 21/4/08 | \n[[8]] | \n[[9]] | \n[[10]] | \n* | \n* | \n
| Seasonal Mean | \n[[11]] | \n[[12]] | \n[[13]] | \n[[14]] | \n[[15]] | \n
Input all entries to 2 decimal places.
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"marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sds1", "maxValue": "sds1", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sdm2", "maxValue": "sdm2", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sdt2", "maxValue": "sdt2", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sdth2", "maxValue": "sdth2", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sdf2", "maxValue": "sdf2", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sds2", "maxValue": "sds2", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sdm3", "maxValue": "sdm3", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sdt3", "maxValue": "sdt3", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sdth3", "maxValue": "sdth3", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sam", "maxValue": "sam", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sat", "maxValue": "sat", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sath-tol", "maxValue": "sath+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "saf", "maxValue": "saf", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sas", "maxValue": "sas", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "Calculate the (unadjusted) seasonal effects for each day of the week: (input all your answers to 2 decimal places).
\nSeasonal effect for Monday: $S_M=\\;$ [[0]]
\nSeasonal effect for Tuesday: $S_T=\\;$ [[1]]
\nSeasonal effect for Thursday: $S_{Thu}=\\;$ [[2]]
\nSeasonal effect for Friday: $S_F=\\;$ [[3]]
\nSeasonal effect for Saturday: $S_{Sa}=\\;$ [[4]]
\n", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sm-tol", "maxValue": "sm+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "st-tol", "maxValue": "st+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sth-tol", "maxValue": "sth+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sf-tol", "maxValue": "sf+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ss-tol", "maxValue": "ss+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "Calculate the adusted seasonal effects for each day of the week: (input all your answers to 2 decimal places).
\nAdjusted seasonal effect for Monday: $S_M=\\;$ [[0]]
\nAdjusted seasonal effect for Tuesday: $S_T=\\;$ [[1]]
\nAdjusted seasonal effect for Thursday: $S_{Thu}=\\;$ [[2]]
\nAdjusted seasonal effect for Friday: $S_F=\\;$ [[3]]
\nAdjusted seasonal effect for Saturday: $S_{Sa}=\\;$ [[4]]
\n", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "asm-tol", "maxValue": "asm+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ast-tol", "maxValue": "ast+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "asth-tol", "maxValue": "asth+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "asf-tol", "maxValue": "asf+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ass-tol", "maxValue": "ass+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "Use the regression equation in part 2 and the adjusted seasonal effects in part 5 to forecast the number of orders on {thisdate}
\nEstimated orders on {thisdate} = ?[[0]] (input to the nearest whole number)
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "estord-tol1", "maxValue": "estord+tol1", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 2, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "{This} has recently opened for business and is open 5 days a week, Monday, Tuesday, Thursday, Friday and Saturday. The number of orders each day during the first three weeks of business is shown below:
\n{table([[\"7/4/08\",{m1},{t1},{th1},{f1},{s1}],[\"14/4/08\",{m2},{t2},{th2},{f2},{s2}],[\"21/4/08\",{m3},{t3},{th3},{f3},{s3}]],[\" \",\"Mon\",\"Tues\",\"Thur\",\"Fri\",\"Sat\"])}
\n", "tags": ["checked2015", "forecasting", "moving averages", "regression", "seasonal adjustments", "seasonality", "statistics", "time series"], "rulesets": {}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Moving averages, regression and seasonal adjustments.
"}, "advice": "a)
\nThe completed moving average table is as follows:
\n| Mon | Tues | Thur | Fri | Sat | |
|---|---|---|---|---|---|
| 7/4/08 | \n* | \n* | \n$\\var{mth1}$ | \n$\\var{mf1}$ | \n$\\var{ms1}$ | \n
| 14/4/08 | \n$\\var{mm2}$ | \n$\\var{mt2}$ | \n$\\var{mth2}$ | \n$\\var{mf2}$ | \n$\\var{ms2}$ | \n
| 21/4/08 | \n$\\var{mm3}$ | \n$\\var{mt3}$ | \n$\\var{mth3}$ | \n* | \n* | \n
b)
\nThe mean value of $T$ is $\\overline{t}=\\frac{3+4+\\cdots+13}{11}=\\frac{88}{11}=8$.
\nThe mean value of $Y$ is $\\overline{y}=\\frac{\\var{summa}}{11}=\\var{meany}$.
\nWe have:
\n\\[\\begin{align}S_{TY}&=\\sum ty-11\\overline{t}\\overline{y}=\\var{sumty}-11\\times 8 \\times\\var{ meany}\\\\
S_{YY}&=\\sum y^2-11\\overline{y}^2=\\var{sumysquared}-11\\var{meany}^2=\\var{syy}\\\\
S_{TT}&=\\sum t^2-11\\overline{t}^2=814-11\\times 64=110
\\end{align}\\]
Estimate for $\\beta$ is $\\frac{S_{TY}}{S_{TT}}=\\var{testbe}=\\var{estbe}$ to 2 decimal places.
\nEstimate for $\\alpha$ is $\\overline{y}-\\beta\\overline{t}=\\var{meany}-\\var{estbe}\\times\\var{meant}=\\var{testal}=\\var{estal}$ to 2 decimal places.
\nSo the linear regression equation is $Y=\\simplify{{estal}+{estbe}T}+\\epsilon$ coefficients to 2 decimal places.
\nc)
\nThe following table shows the calculated seasonal deviations and seasonal means for each day of the week. These are obtained by taking the moving average data away from the original data on orders. The seasonal means for the days are obtained by taking the means in each column.
\n\n| Mon | Tues | Thur | Fri | Sat | |
|---|---|---|---|---|---|
| 7/4/08 | \n* | \n* | \n$\\var{sdth1}$ | \n$\\var{sdf1}$ | \n$\\var{sds1}$ | \n
| 14/4/08 | \n$\\var{sdm2}$ | \n$\\var{sdt2}$ | \n$\\var{sdth2}$ | \n$\\var{sdf2}$ | \n$\\var{sds2}$ | \n
| 21/4/08 | \n$\\var{sdm3}$ | \n$\\var{sdt3}$ | \n$\\var{sdth3}$ | \n* | \n* | \n
| Seasonal Mean | \n$\\var{sam}$ | \n$\\var{sat}$ | \n$\\var{sath}$ | \n$\\var{saf}$ | \n$\\var{sas}$ | \n
d) The seasonal effects for each day of the week are calculated by first finding the means of all the seasonal deviations found in the last table (not including the seasonal means in the last row). Then you take this away from the seasonal mean for each day.
\nWe find that the mean of the seasonal deviations is:
\n\\[\\simplify[all,!collectNumbers]{({sdth1} + {sdf1} + {sds1} + {sdm2} + {sdt2} + {sdth2} + {sdf2} + {sds2} + {sdm3} + {sdt3} + {sdth3}) / 11} = \\var{om}\\]to 3 decimal places.
\n{comm}
\nSeasonal effect for Monday: $S_M=\\;\\simplify[all,!collectNumbers]{{sam}-{om}}=\\var{sm}$ to 2 decimal places.
\nSeasonal effect for Tuesday: $S_T=\\simplify[all,!collectNumbers]{{sat}-{om}}=\\var{st}$ to 2 decimal places.
\nSeasonal effect for Thursday: $S_{Thu}=\\simplify[all,!collectNumbers]{{sath}-{om}}=\\var{sth}$ to 2 decimal places.
\nSeasonal effect for Friday: $S_F=\\simplify[all,!collectNumbers]{{saf}-{om}}=\\var{sf}$ to 2 decimal places.
\nSeasonal effect for Saturday: $S_{Sa}=\\simplify[all,!collectNumbers]{{sas}-{om}}=\\var{ss}$ to 2 decimal places.
\ne)
\nWe further adjust the seasonal effects for each day by finding the mean of the seasonal effects we have just found and then taking this away from each of the seasonal effects.
\nWe find that the mean of the seasonal deviations is:
\n\\[\\simplify[all,!collectNumbers]{({sm} + {st} + {sth} + {sf} + {ss} ) / 5} = \\var{ms}\\] to 3 decimal places.
\n{comm1}
\nAdjusted seasonal effect for Monday $\\;=\\simplify[all,!collectNumbers]{{sm}-{ms}}=\\var{asm}$ to 2 decimal places.
\nAdjusted seasonal effect for Tuesday$\\;=\\simplify[all,!collectNumbers]{{st}-{ms}}=\\var{ast}$ to 2 decimal places.
\nAdjusted seasonal effect for Thursday$\\;=\\simplify[all,!collectNumbers]{{sth}-{ms}}=\\var{asth}$ to 2 decimal places.
\nAdjusted seasonal effect for Friday$\\;=\\simplify[all,!collectNumbers]{{sf}-{ms}}=\\var{asf}$ to 2 decimal places.
\nAdjusted seasonal effect for Saturday $\\;=\\simplify[all,!collectNumbers]{{ss}-{ms}}=\\var{ass}$ to 2 decimal places.
\nf)
\nWe use the regression equation found above $Y=\\simplify[all,!collectNumbers]{{estal}+{estbe}T}+\\epsilon$ to estimate the number of orders on {thisdate}.
\n\nBut we have to adjust for seasonality using the adjusted seasonal effect found above by adding on the adjusted seasonal effect for {thisday} i.e. $\\var{adj}$.
\nNote that $T=1$ corresponds to 7/4/08 and hence $T=\\var{ti}$ for {thisdate}.
\nSo putting $T=\\var{ti}$ gives $Y=\\simplify[all,!collectNumbers]{{estal}+{estbe}*{ti}+{adj}}=\\var{estal+estbe*ti+adj}=\\var{estord}$ orders to the nearest whole number.
\n\n", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}