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Answer :
Final answer:
To forecast the number of students seeking appointments, we calculate the least squares regression line using the given student data from previous weeks. By finding the slope and intercept after assigning numerical values to the weeks, we create the equation of the trend line, which we then use to predict the number of students for this week.
Explanation:
To forecast this week's number of students seeking appointments with Professor Very Busy using the least squares trend line, we will follow a process similar to the examples provided. First, we need to assign numerical values to represent the weeks, where 6 weeks ago is '1' and last week is '6'.
Then, we calculate the mean of the x-values (the weeks) and the y-values (the number of students). After that, we compute the slope (β) of the trend line using the formula β = Σ((x - μx)(y - μy)) / Σ((x - μx)2), where μx and μy are the means of x and y respectively. The next step is to find the intercept (α) with the formula α = μy - βμx.
With the slope and intercept, we can create the least squares regression line's equation, which is ŵ = α + βx. Finally, we can substitute '7' for this week's x-value to predict the number of students.
It is important to note that to ensure accuracy, we should use the same number of significant figures as the data allows when reporting our regression equation. As we are given whole numbers for the number of students, likely, our final forecast should also be a whole number.
Additionally, this method assumes a linear relationship between the weeks and the number of students, which may not always be accurate, especially if there are underlying factors affecting student appointments that are not captured by the week alone.
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Solution:
Smooth the pattern as well as the arrangement to get the outlook this year.
S(Week 5) = TAF(Week 5) + alpha*(Actual(Week 5) - TAF(Week 5))
= 75 + 0.5(65 - 75) = 70
S(Week 6) = TAF(Week 6) + alpha*(Actual(Week 6) - TAF(Week 6))
= 65 + 0.5(50 - 65) = 57.5
T(Week 6) = T(Week 5) + beta*(TAF(Week 6) - TAF(Week 5) - T(Week5))
= -5 + 0.1(65 - 75 - (-5)) = -5 + (-0.5) = -5.5
Therefore, TAF(Week 7) = S(Week 6) + T(Week 6) = 57.5 + (-5.5) = 52.0
