Did I Get This – State Hypotheses (Proportions)
In each of the following examples, a test for the population proportion (p) is called for. You are asked to select the right null and alternative hypotheses.
Scenario 1: When shirts are made, there can occasionally be defects (such as improper stitching). But too many such defective shirts can be a sign of substandard manufacturing.
Suppose, in the past, your favorite department store has had only one defective shirt per 200 shirts (a prior defective rate of only .005). But you suspect that the store has recently switched to a substandard manufacturer. So you decide to test to see if their overall proportion of defective shirts today is higher.
Suppose that, in a random sample of 200 shirts from the store, you find that 27 of them are defective, for a sample proportion of defective shirts of .135. You want to test whether this is evidence that the store is “guilty” of substandard manufacturing, compared to their prior rate of defective shirts.
Scenario 2: It is a known medical fact that just slightly fewer females than males are born (although the reasons are not completely understood); the known “proper” baseline female birthrate is about 49% females.
In some cultures, male children are traditionally looked on more favorably than female children, and there is concern that the increasing availability of ultrasound may lead to pregnant mothers deciding to abort the fetus if it’s not the culturally “desired” gender. If this is happening, then the proportion of females in those nations would be significantly lower than the proper baseline rate.
To test whether the proportion of females born in India is lower than the proper baseline female birthrate, a study investigates a random sample of 6,500 births from hospital files in India, and finds 44.8% females born among the sample.
Scenario 3: A properly-balanced 6-sided game die should give a 1 in exactly 1/6 (16.7%) of all rolls. A casino wants to test its game die. If the die is not properly balanced one way or another, it could give either too many 1’s or too few 1’s, either of which could be bad.
The casino wants to use the proportion of 1’s to test whether the die is out of balance. So the casino test-rolls the die 60 times and gets a 1 in 9 of the rolls (15%).
This document is linked from Proportions (Introduction & Step 1).