Learning Lab

Consider statements such as

1. Teenagers aged \(13\)-\(15\) spend no more than \(10\) hours a week on Facebook.

2. The average weight of Australian men is the same as it was in \(1990\).

3. Students from private schools have the same mean ATAR score as the Victorian average.

4. The mean winter rainfall for the last \(10\) years is the same as the historical mean.

Our confidence about the probabilities of values drawn from normally distributed populations and sampling distributions enables us to formally test hypotheses (or claims) such as these.

When we perform an ‘experiment’ we know there will be chance variation. For example, if we toss a supposedly fair coin \(100\) times we would not be surprised to obtain \(48\) or \(45\) or perhaps even \(40\) heads. However we would be surprised to obtain only \(5\) heads. If we were testing a coin for ‘fairness’ we might decide beforehand what we would consider a reasonable number of heads. In hypothesis testing ‘reasonable’ is defined as what we could expect \(95\)% (or \(99\)% or \(90\)% etc) of the time.

In a hypothesis test we are concerned to assess how unusual our result is, whether it is reasonable chance variation (obtaining \(45\) heads in \(100\) tosses of a coin) or whether the result is too extreme to be considered chance variation (obtaining \(5\) heads in \(100\) tosses of a coin).

A hypothesis test formalizes the process of deciding whether a result is reasonable.

Steps in a Hypothesis Test

The steps are:

1. State the null and alternative hypotheses
   \(H_{o}:\) \(\overline{x}=\mu\) (the sample mean is the same as the population mean after allowing for chance variation)
   \(H_{a}:\overline{x}\neq\mu\) (the sample mean is not the same as the population mean after allowing for chance variation)

2. Significance level \(\alpha\) is chosen (\(\alpha=0.05\Rightarrow\) we are defining reasonable as what we can expect 95% of the time)

3. Critical values
   Tables or a calculator or a computer are used to find the z-values that corresponds to the chosen significance level. These are called the critical values.

4. Calculate the test statistic.
   This is the standardised difference between the sample mean (calculated from the given data) and the known population mean: \[\begin{align*} z & =\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}. \end{align*}\]

5. Decision: Is the result reasonable if \(H_{o}\) is true?
   Is the test statistic more extreme than the critical value?
   Yes \(\Rightarrow\) Reject \(H_{o},\) No \(\Rightarrow\) Do not reject \(H_{o}\).

6. Conclusion
   There is (if you reject)/is not (if you do not reject) evidence to suggest that… (Paraphrase the information in the question to complete the conclusion).

It is important to note that

• The decision about the null hypothesis is not made with certainty but with a level of confidence that the error in the decision is small (for example 5% if \(\alpha=0.05\))
• The decision relates only to rejecting or not rejecting \(H_{o}\). \(H_{a}\) is not mentioned in the decision, and we DO NOT ACCEPT \(H_{o}\) or \(H_{a}\)
• The steps for hypothesis testing may differ from course to course so check with your program.

Exercises

Your answers should be set out and contain all the steps shown above. A brief outline of the main features is given in the answers to teh following exercises.

1. Repeat the example to decide if there is evidence at the \(10\)% level of significance that attending the bridging program is associated with the change in scores.

2. A random sample of \(36\) soft drinks from vending machines had an average content of \(370\)ml with a standard deviation of \(20\)ml. Test the null hypothesis that \(\mu\) \(=375\) ml against the alternative hypothesis \(\mu\) \(\neq375\)ml at the \(1\)% significance level.

3. A bank manager has historical data that shows over lunchtime Mon – Fri the mean number of customers that come into the bank is \(32\). Accordingly he believes he has no need to change the number of tellers. However a branch survey conducted every lunchtime over eight weeks found that the mean number of customers was \(36\) with a standard deviation of \(8.2\). Conduct a hypothesis test with a \(5\)% level of significance to test whether the mean number of lunchtime customers has changed. What recommendation would you make to the bank manager?

4. The manufacturer of ‘longlast’ batteries claims the mean lifetime of his batteries is \(450\) hours. A consumer interest magazine samples 100 batteries and finds that they have a mean of \(444\) hours with a standard deviation of \(28\) hours. Do the sample data contradict the manufacturers claim? (use \(\alpha\) \(=0.02\))