The bookseller
A year of daily book sales as a discrete uniform, the boundaries stated exactly, and a month's average by the central limit theorem.
Draws on 4. The uniform distribution, 10. Sampling distributions and the CLT
Step 1The question
A bookshop sells between 90 and 100 books a day. The owner wants to know how often a day goes well or badly, and what a month of sales looks like. The lab's data are the 365 days of 2025 (bookshop-sales.csv).
Step 2Choosing the model
Sales are whole numbers, and the histogram is roughly flat from 90 to 100: a discrete uniform on the 11 values from 90 to 100, scipy's randint(90, 101), whose upper end is excluded. The continuous uniform would give every single total a probability of 0 (chapter 4).
A good day
What is the probability of selling 96 books or more on a given day?
The year's data had 142 such days, a share of 0.389, below the model's 0.4545. A year is itself a sample; whether a gap this size could be chance is the kind of question a hypothesis test answers.
A slow day
What is the probability of selling fewer than 92 books on a given day?
Step 3The mean and the spread
The model's mean is 95 books and its standard deviation \(\sqrt{\tfrac{11^2 - 1}{12}} = \sqrt{10} = 3.1623\) books. The year's data have a mean of 94.6 and a standard deviation of 3.082.
Step 4A month of sales
A single day is flat, anything from 90 to 100. The average over 30 days is not: by the CLT it is close to normal, centred on 95 with a standard error of \(\tfrac{3.1623}{\sqrt{30}} = 0.5774\) books.
A weak month
Using the CLT, what is the probability that the average daily sales over 30 days are below 94 books?
randint(90, 101), and the boundary matters: fewer than 92 is \(\tfrac{2}{11}\), at most 92 is \(\tfrac{3}{11}\). A single day is flat; the average of a month is nearly normal, with a standard error of 0.5774 books.