Let \(X_1,X_2,X_3,\ldots\) be a sequence of sample variables for a random variable \(X\) with \(E(X)=\mu\) and \(\var(X)=\sigma^2\text{.}\) Let \(S_n\) denote the variable \(S_n=X_1+X_2+\cdots + X_n\text{,}\) so that we have \(E(S_n)=n\mu\) and \(\var{S_n}=n\sigma^2\text{.}\) Let \(T_n=(S_n-n\mu)/(\sqrt{n}\sigma)\text{.}\) We have
Let \(X\) be a random variable with \(E(X)=\mu\) and \(\var(X)=\sigma^2\text{.}\) Let \(\overline{X}=\frac{1}{n}\sum_{i=1}^n X_i\) be the average of a sample of size \(n\) of \(X\text{.}\) For \(\alpha\) in the range \(0\lt \alpha \lt 1\text{,}\) let \(z_\alpha\) be the value of a standard normal variable such that \(P(|z|\leq z_\alpha)=\alpha \text{.}\) The Central Limit Theorem says that
Equation (7.1) motivates the following procedure for estimating an unknown population parameter \(\mu\) from sample data. Intuitively, \(\overline{X}\) estimates \(\mu\) with an error estimated by \(\sigma_{\overline{X}}\text{.}\) If the population variance \(\sigma^2\) is unknown, we can use the sample SD \(s\) to estimate \(\sigma\text{,}\) and we can use \(s/\sqrt{n}\) to estimate \(\sigma_{\overline{X}}\text{.}\) Inspired by (7.1), we say that the interval
is a \(100z_\alpha\%\) confidence interval for the population mean \(\mu\text{.}\) Notice that the confidence interval is a random quantity, that is, it depends on the sample. Roughly speaking, we estimate that approximately \(100z_\alpha\%\) of all \(100z_\alpha\%\) confident intervals will contain \(\mu\text{,}\) and approximately \(100(1-z_\alpha)\%\) of these intervals will fail to contain \(\mu\text{.}\)
Hypothesis testing is using sample data to assign numbers for how skeptical you should be about certain claims. In this subsection we describe a hypothesis test called the 1-tail \(z\) test. We will illustrate using the following example.
Suppose that you are assigned to check the accuracy of a machine that is supposed to make identical widgets with a weight of \(\mu_0\) grams. Your sample of \(n\) widgets has an average weight of \(\overline{X}\) grams, and your weighings have a sample SD of \(s\) grams. There is a difference between your sample average \(\overline{X}\) and \(\mu_0\text{.}\) How do you decide whether the difference is significant? In a hypothesis test, we play “what-if?”, in the following way. We suppose that the machine actually performs according to the claimed standards, so that the average widget has a weight of \(\mu_0\text{.}\) Now we ask, what is the probability that a random sample of size \(n\) would be as far away or farther from \(\mu_0\) as the value \(\overline{X}\) that we observed? By the Central Limit Theorem, the answer is
If this probability is small, we feel skeptical of the claim that the machine produced widgets with an average weight of \(\mu_0\text{.}\) If this probability is large, we do not feel skeptical. The threshold probability value for skepticism is adjustable. The most common default value is \(0.05\text{.}\) The formal structure of this type of hypothesis test, called a 1-tail \(z\) test is this.
A null hypothesis, denoted \(H_0\text{,}\) is made. This is a claim that a population average for a certain random variable \(X\) has a certain value \(\mu_0\text{.}\)
A conclusion is made: if \(P\lt \alpha\text{,}\) the data is called significant. The null hypothesis is rejected. If \(P\gt \alpha\text{,}\) the data is called not significant. The null hypothesis is not rejected.
