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The British government carries out a census of the entire population of the United Kingdom every 10 years (most recently in April 2001).
The first census in the United Kingdom was carried out in 1086 with the construction of the Doomesday Book. However they have only been conducted on a regular basis since 1801.
The census provides the government with a detailed picture of the population living in each part of the country (town, city or countryside). The results are used to help plan public services (health, housing, transport and education) for the future.
Carrying out a census of the entire population is usually not feasible or sensible.
Introduction to sampling
money
time
resources
In addition, some investigations could result in the destruction of the entire population!
For example, if a light bulb manufacturer wished to investigate the lifetime of its bulbs, a census would result in the destruction of all the bulbs it produced.
Instead of surveying the whole population, information can instead be obtained from a sample. The sampling process should be undertaken carefully to ensure that the sample is representative of the entire population. Bias can occur if one section of the population is over- or under-represented.
Introduction to sampling
Question: A local council wishes to know the views of local people on public transport. Criticize each of the following sampling regimes:
1. Ask the people waiting at the town centre bus stop.
2. Leave questionnaires in local libraries for people to fill in.
3. Ask people at the shopping centre on a Thursday morning.
One way to obtain a fair sample is to use random sampling. This method gives every member of the population an equal chance of being chosen for the sample.A more formal definition of a random sample is as follows:
There are a number of ways in which a random sample can be chosen. One commonly used technique is to use random number tables.
Sampling methods
A sample of size n is called a random sample if every possible selection of size n has the same probability of being chosen.
Here is how to use random digits to obtain a sample:
Random number tables
Example: A sample of size 15 is required from a population of size 300.
One possible approach would be to obtain a sampling frame for the population and number every member from 001 to 300. You could then obtain chains of 3 random digits from tables. If the chain corresponds to a number between 001 and 300 you could select that member of the population; otherwise you could discard that chain and choose another.
So, we noticed from the previous slide that, with 20 throws of a fair die, the probability of getting 7 or more sixes is about 0.0371.
This means that if a fair die were thrown 20 times over and over again, then you would obtain 7 or more sixes less than once in every 20 experiments.
The figure of 1 in 20 (or 5%) is often taken as a cut-off point – results with probabilities below this level are sometimes regarded as being unlikely to have occurred by chance.
However, in situations where more evidence is required, cut-off values of 1% or 0.1% are typically used.
The null hypothesis (H0) is often thought of as the cautious hypothesis – it represents the usual state of affairs.
The alternative hypothesis (H1) is usually the one that we suspect or hope to be true.
Hypothesis testing is concerned with examining the data collected in experiments, and deciding how likely the result is to have occurred if the null hypothesis is true.
The significance level of the test is the chosen cut-off value between the results that might plausibly have been obtained by chance if H0 is true, and the results that are unlikely to have occurred.
Significance levels that are typically used are 10%, 5%, 1% and 0.1%.
These significance levels correspond to different rigours of test – the lower the significance level, the stronger the evidence the test will provide.
A formal introduction to hypothesis tests
Note: It is important to appreciate that it is not possible to prove that a hypothesis is definitely true in statistics. Hypothesis tests can only provide different degrees of
evidence in support of a hypothesis. A 10% significance level can only provide weak evidence in support of a hypothesis. A 0.1% test is much more
Do you think you can taste the difference between branded chocolate and supermarket own-label chocolate?
You are going to perform an experiment to find out.
There will be 2 pieces of chocolate to try: one will be a branded make of chocolate, the other will be a supermarket’s own-brand. Try to identify the branded make.
The significance level in this test was chosen to be 5% – the probability calculated was much higher than this.
We conclude: the evidence is not strong enough to reject H0 at the 5% significance level. The data does not provide evidence that Mr Jones was exaggerating his support.
Examination style question: The standard treatment for a particular medical condition has a success rate of 70%. A new drug is launched which, it is claimed, treats a greater proportion of patients successfully.
A doctor tries the new drug on 20 patients and finds that it successfully treats 19 of them.
Test at the 1% significance level whether there is evidence to suggest that the new drug treatment is more successful than the standard treatment.
The examples considered so far can all be classified as one-sided tests – we have been testing for either an increase or a decrease in the value of the parameter, p.
Sometimes we are not looking specifically for an increase (or decrease) in p, but instead we may want to examine whether the value of p has changed. In these situations we use a two-sided (or a two-tailed) test.
A two-sided hypothesis test carried out at the α% significance level is in a sense two separate one-sided tests. The significance level is therefore shared between these two tests, ½α% for each tail.
Example: A restaurant has traditionally found that 60% of its customers have been pleased or very pleased with the quality of the food served.
A new chef is appointed and the restaurant management wish to find out whether this has changed the proportion of customers who are happy with their food.
The management question 16 diners and discover that 14 of them are pleased or very pleased with their food.
Test at the 5% significance level whether there has been a change in the proportion of contented customers.
If H0 were true, we would expect 16 × 0.6 = 9.6 customers to be pleased with the food quality. The observed number, 14, is on the high side.
We calculate P(X ≥ 14):
One-sided versus two-sided tests
( ) . . .
( ) . . .
( ) . .
16 14 214
16 15 115
16
P 14 0 6 0 4 0 0150
P 15 0 6 0 4 0 0030
P 16 0 6 0 0003
X C
X C
X
So P(X ≥ 14) = 0.0183 < 2.5%.
Conclusion: We can reject the null hypothesis at the 5% significance level. There is some evidence that the proportion pleased or very pleased with their food has changed.
Examination style question: A driving instructor knows from past experience that 2 out of 3 of his students pass their driving test first time.
A new driving examiner is employed at the test centre. The instructor wants to know whether this has changed the proportion of his students passing their test at the first attempt.
He monitors the next 12 of his students taking their tests and finds that 6 pass their test first time round.
One-sided versus two-sided tests
a) Write down a suitable null and alternative hypothesis for this test. Explain why your alternative hypothesis has the form it has.
b) Carry out the test at a 10% significance level.
We would expect 8 candidates to pass on the first attempt if the null hypothesis were true. The observed number, 6, is on the low side.
We need to calculate P(X ≤ 6).
Using tables, this probability is 0.1777 > 5%.
Conclusion: We are unable to reject the null hypothesis. The data does not provide enough evidence to suggest that the proportion of candidates passing their driving test at the first attempt has altered.
Example 1: Police records show that 25% of the vehicles using a stretch of road exceed the speed limit. A new speed camera is installed. The police wish to find out whether this has led to a reduction in the proportion of drivers speeding.
The police sample 20 cars driving along the stretch of road.
Critical regions
The critical (or rejection) region for a hypothesis test is the range of values for which the null hypothesis could be rejected.
a) Find the critical region for a test carried out at the 5% significance level.
b) Comment on the implications of the test if the police find 2 speeding drivers.
This number is not contained within the critical region.
Therefore we cannot reject the null hypothesis at the 5% level. The evidence does not support the theory that the proportion of motorists that speed has changed.
Examination style question: A gardener knows from past experience that 80% of the runner bean seeds that he plants will germinate. He is forced to switch to a different brand of seed. He wants to find out whether this has led to a change in the germination rate of his runner beans.
He plants 25 seeds. Let X represent the number of seeds that germinate.
Find the critical region for a hypothesis test carried out at the 10% significance level.
A car salesman sells on average 2 new cars every day.
His company asks him to change his sales strategy.
The salesman records how many cars he sells over the next 7 days so that he can test whether there has been any change in how successfully he sells new cars.
Find the critical region for a hypothesis test using a nominal 5% significance level. The probability of rejection in each tail should be as close as possible to 2.5%.
Remember that the critical region for a hypothesis test is the set of values that would lead to the rejection of the null hypothesis.