Statistical Inference for the Mean Objectives: (Chapter 8&9, DeCoursey) -To understand the terms variance and standard error of a sample mean, Null Hypothesis, Rejection Region, Type I and II errors, and confidence interval. -To understand how to perform hypothesis testing for a population and a sample (t-test).

Statistical Inference for the Mean Variance of a sample mean: an indication of the reliability of the mean as an estimate of the population mean. A sample consists of n independent observations. The mean of the observations Let the first observation comes from a population of variance σ 1 2, and so on to the nth observation. Then the variance of is

Statistical Inference for the Mean Variance of a sample mean: If the variables all come from the same distribution with variance σ 2 Standard error of a sample mean: (example)

Statistical Inference for the Mean Statistical Inference is a process of using observations of a sample to estimate the parameters of the population. Statistical Inference for the Mean: To provide answers on whether the sample mean and the mean of the corresponding population is close enough together so that it is reasonable to say that the sample might have come from the population. This is done by assuming the variance of the population is known and not changed.

Statistical Inference for the Mean E.g. When a manufacturing process is operating properly, the mean length of a steel rod is known to be inches, and lengths are normally distributed. The standard deviation of this length is inches. If a sample consisting of 6 items taken from current production has a mean length of inches, is there evidence that the mean of the population is changed? Or the sample is from different population so that some adjustment of the process is required?

Statistical Inference for the Mean Null Hypothesis: the notion or suggestion that nothing has changed. E.g. “the population mean is still inches.” If the probability of the observation “mean length of inches” is very small based on the given population mean and variance, i.e. near the tails, there is a potential to reject the null hypothesis.

Statistical Inference for the Mean We are interested in the probability that the variable takes on a value of 2.14 standard deviates from the mean in either direction. (Two-tailed test) μ

Statistical Inference for the Mean Should we reject the Null Hypothesis? μ

Statistical Inference for the Mean Then has to be defined. Reject region: the region of a probability distribution where the probability of the observation is very small, i.e. near the tails μ

Statistical Inference for the Mean The low probability of the observation suggests that something has changed. (i.e. the chances of it occurring randomly are small) and we must reject the Null Hypothesis. Reject region = e.g. level of significance=5% When the probability of the observation called observed level of significance (3.2%) is Less than 5%, reject the Null Hypothesis; Otherwise do not reject μ

Statistical Inference for the Mean E.g. When a manufacturing process is operating properly, the mean length of a steel rod is known to be inches, and lengths are normally distributed. The standard deviation of this length is inches. If a sample consisting of 6 items taken from current production has a mean length of inches, is there evidence at the 5% level of significance that some adjustment of the process is required? Answer:

Statistical Inference for the Mean Type I error is to reject the null hypothesis when it is true. In the case of a mean, this occurs when the null hypothesis is correct, but an observation or sample mean is so far from the expected mean by chance that the null hypothesis is rejected. We may decide to take some action on the basis of the test of significance. But we can never be completely certain we are taking the right action. The probability of a Type I error is equal to the level of significance. To reduce the chance of a type I error: Small rejection region (small level of significance).

Statistical Inference for the Mean Type II error is to accept Null Hypothesis when it is False. If in fact the population mean has changed, the null hypothesis is false. But the sample mean might still by chance come close enough to the original sample mean so that we would accept the null hypothesis, giving a Type II error. The probability of a Type II error depends on how much the population mean changed in comparison to the standard error of the mean. Reduce the chance of a type II error: larger rejection region.

Statistical Inference for the Mean How to perform a statistical test for the mean when σ of the population is known? - State the null hypothesis in terms of a population parameter, such as mean μ. - State the alternative hypothesis in terms of the same population parameter. - Assume the variance is known as well as the mean. - Calculate the test statistic z of the observation using the mean and variance given by the null hypothesis. - State the level of significance – rejection limit. - If probability falls outside of the rejection limit, we reject the Null Hypothesis, which means the measurement is from a different population. We are assuming Normal Distribution in this scenario.