Question

The president of a small manufacturing firm is concerned about the continual increase in manufacturing costs over the past several years. The following figures provide a time series of the cost per unit for the firm's leading product over the past eight years.
Year Cost/Unit ($) Year Cost/Unit ($)

1 20.00 6 30.00
2 24.50 7 31.00
328.20 8 36.00
4 27.50
5 26.60

(a) Make the correct time series plot. What type of pattern exists in the data?

(b) Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series. If required, round your answers to two decimal places.

y-intercept, b0 =

Slope, b1 =

MSE =

(c) What is the average cost increase that the firm has been realizing per year? Round your interim computations and final answer to two decimal places. $

(d) Compute an estimate of the cost/unit for next year. If required, round your answer to two decimal places.

Answer 1

Answer:

The answers to the question are

(a) The time series plot is given as attached

(b) The parameters for the line that minimizes MSE for the time series are;

y-intercept, b₀ = 19.993

Slope, b₁ =  1.77

MSE = 19.44

T$_t$ = 19.993  + 1.774·t

(c) The average cost increase that the firm is realizing per year is $ 1.77

(d) The estimate of the cost/unit for next year is $35.96.

Step-by-step explanation:

(a) Using the provided data, the time series plot is given as attached

(b) Given hat the y-intercept, = b₀

Slope = b₁

Therefore the  linear trend forecast equation is given s

T$_t$ = b₀ + b₁·t

The linear trend line slope is given as

b₁ = $\frac{\Sigma^n_{t=1}(t-\overline{\rm t)}(Y_t-\overline{\rm Y)}}{\Sigma^n_{t=1}(t-\overline{\rm t} )^2}$

b₀ = $\overline{\rm Y}$ - b₁·$\overline{\rm t}$

Where:

Y$_t$ = Time series plot value at t

n =  Time period number

$\overline{\rm Y}$ = Time series data average value and

$\overline{\rm t}$ = Average time, t

Therefore, $\overline{\rm t}$ = $\frac{\Sigma^n _{t=1} t}{n} = \frac{36}{8} =4.5$

$\overline{\rm t}$ = 4.5

$\overline{\rm Y}$  =  $\frac{\Sigma^n _{t=1} Y_t}{n} = \frac{223.8}{8} =27.975$

$\overline{\rm Y}$  =  27.975

Therefore the linear trend line equation T$_t$, is

b₁ = $\frac{\Sigma^n_{t=1}(t-\overline{\rm t)}(Y_t-\overline{\rm Y)}}{\Sigma^n_{t=1}(t-\overline{\rm t} )^2}$ =  $= \frac{74.5}{42}$ = 1.774

b₀ = $\overline{\rm Y}$ - b₁·$\overline{\rm t}$ = 27.975 - 1.774×4.5 = 19.993

Therefore the trend equation for the linear trend is

T$_t$ = 19.993  + 1.774·t

MSE = $\frac{1}{2} \Sigma(Y-\overline{\rm Y)^ }^2$ = $\frac{155.495}{8}$ = 19.44

(c) From the linear trend equation, the average is given as the slope of the curve or b₁ which is equal to 1.774

Therefore the average cost increase that the firm has been realizing per year is $ 1.77

(b) From the equation of the future trend, we have when y = 9

T$_t$  is given as

T$_t$ = 19.993  + 1.774×9 = 35.96

The cost/unit for 9th year is $35.96