website/content/posts/estimating-delivery-time.md

title	date	tags
Estimating package delivery time	2022-06-21T18:44:53+02:00	math statistics

In the last few months I have ordered a dozen of items from Japan. In total there were six distinct orders that were delivered to me in separate boxes. Here's the data for the time between being sent out (dispatch from outward office of exchange) from Japan and arriving (arrival at inward office of exchange) in Poland.

Location	Outbound	Inbound	Diff days
TOKYO INT BAG	3/29/2022 4:20	4/3/2022 6:22	5.084722222
OSAKA INT BAG	4/9/2022 14:30	4/15/2022 6:38	5.672222222
OSAKA INT BAG	4/11/2022 14:30	4/22/2022 5:29	10.62430556
TOKYO INT BAG	4/19/2022 13:20	4/22/2022 6:05	2.697916667
OSAKA INT BAG	4/21/2022 14:30	4/28/2022 18:33	7.16875
TOKYO INT BAG	5/10/2022 13:20	5/18/2022 2:37	7.553472222

We have n = 6 samples and using this data we can calculate the mean \bar X as well as the standard deviation s of the sample.

$$ \bar X \approx 6.466898148 $$ $$ s \approx 2.672243145

We do not know the properties of the population, and we only have 6 samples, so we have to use the Student's t-distribution here. There are k = n - 1 = 5 degrees of freedom.

Let's assume confidence level of 95%. $$ \alpha = 1 - 0.95 = 0.05

We can use the table for t-distribution to find that the value of t_{\alpha, n-1} = 2.571. Now we can calculate the confidence interval for the mean \mu delivery time between Japan and Poland.

$$ (\bar X - t_{\alpha, n-1} \frac{s}{\sqrt{n}}; \bar X + t_{\alpha, n-1} \frac{s}{\sqrt{n}}) $$ $$ t_{\alpha, n-1} \frac{s}{\sqrt{n}} \approx 2.804347194 $$ $$ (\bar X - 2.804347194; \bar X + 2.804347194) = (3.662550954; 9.271245342)

With these results we can say with 95% confidence that the mean time of delivery from Japan to Poland lies somewhere between 3.66 days and 9.27 days. We can also calculate different results for different confidence levels to get some more interesting overview:

Confidence level	t_{\alpha, n-1} \frac{s}{\sqrt{n}}	Confidence interval
99%	4.398820806	(2.068077342; 10.86571895)
95%	2.804347194	(3.662550954; 9.271245342)
90%	2.198294244	(4.268603904; 8.665192392)
80%	1.610099019	(4.856799129; 8.076997167)
50%	0.792770797	(5.674127351; 7.259668946)