Statistical Inference

1. Statistical Inference Statistical inference is concerned with the use of sample data to make inferences about unknown population parameters. For example, suppose we have a random variable X which has a normal distribution with unknown mean and variance. Examples of statistical inference are hypothesis tests about the population mean or the derivation of intervals within which the mean is likely to lie.

2. Hypothesis Tests • To conduct a hypothesis test we need the following: • A hypothesis to be tested (usually described as the null • hypothesis) and an alternative against which it can be tested. • 2. A test statistic whose distribution is known under the null • hypothesis. • 3. A decision rule which tells us when to reject the null • hypothesis and when not to reject it.

3. Hypotheses can either have one-sided or two-sided alternatives. Two-sided alternatives lead to two-tailed tests while one- sided alternative lead to one-tailed tests.

4. The following outcomes are possible when we perform a test: When we choose our decision rule we would like to avoid making errors of either kind.

5. It is relatively easy to determine the probability of making a Type I error. We choose a critical value which determines a particular probability of falsely rejecting the null. One-tailed test Two-tailed test This is known as the size of the test. Note that the smaller we make the probability of a Type I error then the larger is the probability of a Type II error.

6. The power of a test is defined as: Therefore the stricter is the test (i.e. the smaller is its size) then the lower is its power. It is hard to attach a specific number to the power of a test but we can often rank different tests in terms of their relative power. If a number of different tests are available then we would normally choose the most powerful test.

7. P-Values When deciding whether or not to reject the null hypothesis, the following are often considered. The P-Value is the probability of obtaining a result at least as extreme as the test statistic under the assumption that the null is true. The p-value tells us the probability that we will be making a Type I error if we reject the null. For example, suppose we have a variable which is assumed to follow a standard normal distribution under the null. If we obtain a test statistic of 1.5 then the p-value is 0.067 for a 1-tailed test and 0.134 for a 2-tailed test.

8. Critical Values A critical value is a value corresponding to a predetermined p-value. For example a 5% critical value is a value of the test statistic which would yield a p-value of 0.05. Critical values are often set at 10%, 5% or 1% levels. For example, for the standard normal distribution critical values for a 1-tailed test are: 10% 1.282 5% 1.645 1% 2.326 If the test statistic exceeds the critical value then the test is said to reject the null hypothesis at that particular critical value.

9. Confidence Intervals The confidence interval is an interval estimate of a parameter of interest. It indicates an interval within which the unknown parameter is likely to lie. Confidence intervals are particularly associated with classical or frequentist statistical theory. A confidence interval is normally expressed in percentage terms e.g. we are ’95% confident that the population mean lies between the limits L and U’. The way to interpret a 95% confidence interval is that if we were to repeat a random experiment 100 times then 95 of the interval estimates we calculate would contain the true population parameter.

10. Example: Suppose we have a random variable X which follows a normal distribution. We wish to test: We have a sample of N observations which we use to calculate the sample mean and the sample standard deviation. Under the null hypothesis the following random variable is N(0,1) We can calculate this for a particular sample mean and compare it with a critical value from the t tables to decide if we should reject the null.

11. Alternatively we can use the z transformation to derive a confidence interval. This allows us to derive a confidence interval for the true population mean.

12. Tests based on the t-distribution The problem with the test statistics we have looked at so far is that they depend on an unknown population variance. If we replace the unknown population variance with the sample variance then the resulting test statistic can be shown to follow a student’s t distribution. This is an operational test statistic because we can evaluate it for particular values of the sample mean and the sample standard deviation.

13. Tests based on the chi-squared distribution Chi-squared tests can be used to test if the standard deviation is equal to a particular value: e.g. suppose we wish to test Under the null hypothesis the following test statistic will follow a chi-squared distribution with N-1 degrees of freedom (where n is the sample size).

14. Tests based on the F-distribution The F-test can be used when we wish to test if the variance or the standard deviation has changed. For example, suppose we estimate the standard deviation based on n1 data points. We then observe another n2 data points and wish to test: Under the null hypothesis we have a test statistic of the form:

15. There is an interesting relationship between the t distribution and the F distribution. Let It follows that: We can therefore think of the t distribution as a special case of the F distribution and, in this special case, we can perform either a t test or an F test with identical results.