Testing the mean

Seller Door is a door-to-door sales company. Salespeople for the company have always had a particular routine for selling their products to a potential customer. This routine has been doing well, with a reasonable success rate. But the company recently changed the routine for the salespeople, in the hope of improving this success rate.

But has there been a change in the length of time it takes to go through the routine? Extensive research has shown that the old routine took an average of just over ten minutes: 620 seconds to be exact. With any luck, the new routine is faster. Or perhaps, in an effort to boost the rate of sales, the routine has slowed? Or maybe it hasn't changed.

The company decides to conduct a hypothesis test for the mean μ of the variable X, the time taken for a salesperson to complete a sales routine for a customer. This will be a two-sided test, because the company doesn't have any evidence or reason to suggest that the mean differs from the old value (of 620 seconds) in a particular direction. It simply wants to test if it differs at all from this old value.

So the null and alternative hypotheses for this test will be:

H₀: μ = 620

H_A: μ ≠ 620

We will now spend some time looking at how the company might go about conducting this hypothesis test. We will go through the test twice. First, we will assume that the population standard deviation, σ, is known. Then, we will conduct the test without making this assumption to see what effect this has on the method.