Inference on Mean

The observed values $X_1,\ldots,X_n$ of size

are also known as data, and regarded as independent random variables governed by an underlying probability distribution. Furthermore, it is often assumed that the underlying distribution is a normal distribution with $(\mu,\sigma^2)$ . The mean $\mu$ and the standard deviation $\sigma$ are unknown and called parameters. An estimate of parameter $\mu$ is a “best guess” of the true value from data, denoted by $\hat{\mu}$ . The sample mean $\bar{X}$ is in some sense a best guess of the mean parameter $\mu$ .

Standard error (SE). A random variable constructed from the data $X_1,\ldots,X_n$ is called a statistic, and it remains random until it was observed. Thus, the estimate $\hat{\mu}$ is a statistic. Moreover, it is normally distributed with the mean $\mu$ and the standard deviation $\sigma/\sqrt{n}$ . Since the sample standard deviation is the estimate for $\sigma$ , the statistic $SE(\hat{\mu}) = S/\sqrt{n}$ estimates the standard deviation of $\hat{\mu}$ , and it is called the standard error (SE). Then the margin of error for the estimate $\hat{\mu}$ is calculated along with critical value from t-distribution; see Confidence Interval.

Neyman-Pearson framework. The process of determining “yes” or “no” from the outcome of experiment is called a hypothesis test . A widely used formalization of this procedure is due to Neyman and Pearson. Suppose that a researcher is interested in whether a new drug works. Then null hypothesis may be that the drug has no effect —it is often the reverse of what he or she actually believe, why? Because the researcher hopes to reject the hypothesis and announce that the new drug leads to significant improvements. If the null hypothesis is not rejected, the researcher announces nothing and goes on to a new experiment.