**4.1 Using null and alternative hypotheses**

**Is some desired effect present in the population or not?**

With this type of question we want to know whether or not a factor or a treatment has an effect on a population. Such questions are answered by conducting research that uses available evidence, observations and experimentation.

Have a look at the issue regarding Vitamin C.

You may believe or have heard people say that taking large doses of vitamin C prevents colds. Vitamin C is a nutrient that is essential to humans for normal functioning. However, some scientists postulated that Vitamin C also could help people to fight off colds and flu. This idea has often been attributed to the Nobel Prize winner for Chemistry, Dr. Linus Pauling.

So the question is: "Does Vitamin C have no effect on colds in the population or does it have an effect?"

The question is answered by developing null and alternative hypotheses.

**EXAMPLE**

In Toronto, an experiment was conducted to discover if large doses of vitamin C prevented colds (Moore 1991). 500 volunteers were randomly assigned, half each, to the treatment group and the control group. Members of the treatment group were given 1g/day of vitamin C (about one vitamin C tablet/day) and if they felt as though they might be getting a cold the dosage was increased to 4g/day. Those in the control group were given a placebo.

It is possible for this study to be framed in a particular way that indicates competing beliefs about the population.

- First we assume that Vitamin C makes no difference - has no effect. This is called the null hypothesis - the no change hypothesis. The symbol used is H
_{0}- "H" for hypothesis and "0" for zero change. (The word "null" is another way of saying "zero".) - Second we set up an alternate hypothesis H
_{a}(or H_{1}) which takes the opposite point of view - namely Vitamin C does make a difference. - The data is used to produce a test value - a test statistic - in this case it measures the average number of colds caught by each group (treatment group and control group).
- Then the process evaluates whether the difference between the groups is significantly large.

So, in statistical inference the language used is to talk about *alternative hypotheses*. The current belief (no change situation) is written as a hypothesis called a null hypothesis (H_{0}) and the alternative position is written as a hypothesis called the alternative hypothesis (H_{a} or H_{1}).

We can write the hypotheses as follows:

H | there is no difference between the Vitamin C group and the placebo group |

H | there is a significant difference between the Vitamin C group and the placebo group |

H | the Vitamin C group and the placebo group catch the (approximately) same number of colds |

H | the Vitamin C group catches a significantly smaller number of colds |

or, using symbols, where µ_{C} is the mean of the population taking Vitamin C and µ_{N} is the mean of the population not taking Vitamin C (control group) :

H

_{0}:µ

_{C}= µ_{N}H

_{a}:µ

_{C}? µ_{N}

or:

H

_{0}:µ

_{C}= µ_{N}H

_{a}:µ

_{C}< µ_{N}

From the data, an appropriate test statistic can be calculated that assesses the strength of the evidence against the null hypothesis. If the statistic is significant then the data lead you to reject the null hypothesis and accept the alternative hypothesis. [This process is beyond the scope of our module.]

To be statistically significant the experimental evidence for Vitamin C needs to be so strong that it would appear by chance only rarely in many experiments.

That is, if there was no significant difference between Vitamin C and the placebo, the only differences occurring in the mean number of colds caught by each group would be due to random effects.

*EXAMPLES*

- The General Manager of Fresh'n'Easy air conditioners tells an investigative reporter that at least 85% of its customers are "completely satisfied" with their overall purchase performance. What hypotheses will be used by the reporter to test the claim?

Answer:

Let π be the true proportion of "complete satisfaction" in the population.

H | there is no difference between the manager's claim and reality, |

H | less than 85% of customers are satisfied |

H | π = 0.85 |

H | π < 0.85 |

- A consumers' advocate claims that the contents of only 70% of muesli packets produced by Big Oats have contents with the same mass as claimed on the packet. What hypotheses will be used to test the claim?

Answer:

Let π be the true proportion of packets in the population which weigh the same as the stated contents.

H | there is no difference between the advocate's claim and reality, |

H | 70% do not match the stated contents |

H | = 0.70 |

H | π ? 0.70 |

- A student counsellor claims that first year Science students spend an average 3 hours per week doing exercises in each subject. What hypotheses will be used by a lecturer to test the claim?

Answer:

Let µ be the true mean number of hours spent per week

H | there is no difference between the counsellor's claim and reality, |

H | the amount of exercise hours is not 3 (it could be more, it could be less) |

H | µ =3 |

H | µ ? 3 |

- This example deals with credit accounts held with a company. To conclude that the system is cost effective requires the manager to show that the mean account balance for all customers is greater than $1700 - so we set up hypotheses:

Answer:

Let µ be the true mean account balance

H | the system is not effective |

H | the system is effective |

H | µ = 1700 |

H | µ > 1700 |

### Questions

Which is the null hypothesis and which is the alternative hypothesis in each of the following situations?

### Question 1

A vaccine made from tumour cells was administered to people with malignant melanoma - a deadly form of skin cancer. The vaccine was given to people whose melanoma had spread. The results showed that 27% survived for 5 years compared with 10% survival rate over the same time period for those receiving other treatments.

- The vaccine is more effective in increasing the life expectancy of people with malignant melanoma than other treatments.
- The vaccine has the same effect on the melanoma as a standard treatment.

### Question 2

A study was conducted to find out if zinc supplements reduced the severity of colds and influenza compared with not taking anything.

- Taking zinc supplements does not effect the severity of colds and influenza.
- Taking zinc suplements reduces the severity of colds and influenza.

### Question 3

A sociologist wants to find out if more Liberal voters than Labour favour the "Three strikes and you're out" rules for juvenile crime offenders (if juvenile offenders are found responsible for a third criminal offence, they are sent to adult gaol rather than a juvenile facility).

- Liberal voters are more likely than labour voters to favour the "Three strikes and you're out" rule.
- With respect to their views about the "Three strikes and you're out" rule, there is no difference between liberal and labour voters.