Data Analysis A Gentle Introduction – Graham Upton

Figure 6.1 illustrates two cases where the p-value is 0.05. In the two-sided case there is a probability of 0.025 in each tail. In the one-sided case the full 0.05 occurs in one tail only. 6.2.2 Interpreting p-values A rough interpretation of a p-value might be as follows: p-value Interpretation 0.01 < p ≤0.05 There is some evidence that contradicts the null hypothesis.

Ideally we would like more data to provide confirmation. 0.001 < p ≤0.01 There is strong evidence to reject the null hypothesis. p ≤0.001 There is overwhelming evidence against the null hypothesis. To be clear, the case p ≤0.001 is stating that if the null hypothesis were true, and we were to look at a thousand situations similar to that currently studied, then perhaps only one of those would have results as extreme as has occurred in our case.

Example 6.2 Asanexample,supposewetestthehypothesisthatapopulationhasmean5,usinga randomsampleof100observations.Figure6.2showsahistogramofthesampledata (which was a random sample from a normal distribution with mean 6 and standard deviation 2). 6.2 p-values and the null hypothesis The sample mean is 6.1. To assess whether this is unreasonably far from the hypothesized mean of 5, we use a so-called t-test. This makes use of the t-distribution, which is a symmetric distribution very similar to the normal distribution.

Frequency –2 Sample values Figure 6.2 Histogram for a sample of 100 observations from a population with mean 6. The hypothesized mean was 5. The output from R states that the p-value is ‘6.955e-05’, which is a shorthand for 0.00006955. In other words, if the mean were 5, then the probability of rolling a sam- ple mean this different, or more different, from the hypothesized value is less than 7 in 100,000.

The null hypothesis would most certainly be rejected: we would con- clude that the true population mean was around 6.1 (since that was the sample mean).

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DOI: 10.1093/oso/9780192885777.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. Contents Preface ix 1.

First steps 1 1.1 Types of data 1 1.2 Sample and population 2 1.2.1 Observations and random variables 3 1.2.2 Sampling variation 3 1.3 Methods for sampling a population 4 1.3.1 The simple random sample 5 1.3.2 Cluster sampling 5 1.3.3 Stratified sampling 6 1.3.4 Systematic sampling 7 1.4 Oversampling and the use of weights 7 2.

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