Question

Judy manages a large used car dealership that has experienced a steady growth in sales during the last few years.  Using time series regression and sales data for the last 20 quarters, Judy obtained a forecast of 800 car sales for next quarter.  With her model and the past data the standard error of the forecast was 50 cars.  What are the limits for a 95% forecast range?  for an 80% forecast range?

Answer 1

Solution

Back-up Theory

Given X ~ N(μ, σ²), 100(1 - α) % Confidence Interval for μ, when σ is known:

Xbar ± (Z_{α /2})σ/√n   where ………………………………………………………………………… (1)

Xbar = sample mean, Z_{α /2} = upper (α /2)% point of N(0, 1), σ = population standard deviation and n = sample size.

Now to work out the solution,

Given Xbar = 800, σ = 50 and n = 20.

Part (a)

95% forecast range => α = 0.05 i.e., 5% and hence

Z_{α /2} = 1.96 [Using Excel Function: Statistical NORMSINV].   So, vide (1),

95% forecast range is: 800 ± (1.96 x 50/√20)

= 800 ± 21.91

= [778.1, 821.9]   Answer

Part (b)

Here, α = 0.2 i.e., 20% and hence

Z_{α /2} = 1.2816 [Using Excel Function: Statistical NORMSINV].   So, vide (1),

80% forecast range is: 800 ± (1.2816 x 50/√20)

= 800 ± 14.33

= [785.7, 814.3] Answer

DONE