Asssessing slope from a single data set
Use this page to show that the randomness of the LS slope implies uncertainty about whether β1 is zero when b1 is close to zero.
Change the slope to zero then take a few samples.
The least squares slope is usually non-zero, so a single data set cannot immediately tell you whether β1 is zero.
The context of the model is the price of a popular New Zealand cabernet sauvignon red wine and its effect on sales in a supermarket chain, measured as a proportion of total red wine sales in a week. The relationship between price and sales will be nonlinear at high prices, but is expected to be reasonably linear within a price range of $12 to $20 per bottle.