9.3 Distribution needed for hypothesis testing

Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing.We will perform hypotheses tests of a population mean using a normal distribution or a Student's t -distribution . (Remember, use a Student's t -distribution when the population standard deviation is unknown and the sample size is small, where small is considered to be less than 30 observations.) We perform tests of a population proportion using a normal distribution when we can assume that the distribution is normally distributed. We consider this to be true if the sample proportion, $p\text{'}$ , times the sample size is greater than 5 and 1- $p\text{'}$ times the sample size is also greater then 5. This is the same rule of thumb we used when developing the formula for the confidence interval for a population proportion.

Hypothesis test for the mean

Going back to the standardizing formula we can derive the test statistic for testing hypotheses concerning means.

The standardizing formula can not be solved as it is because we do not have μ, the population mean. However, if we substitute in the hypothesized value of the mean, μ ₀ in the formula, we can compute a Z value. This is the test statistic for a test of hypothesis for a mean and is presented in [link] . We interpret this Z value as the associated probability that a sample with a sample mean of $\bar{X}$ could have come from a distribution with a population mean of H ₀ and we call this Z value Z _c for “calculated”. [link] shows this process.

In [link] the three possible outcomes are presented as ${\bar{X}}_1$ , ${\bar{X}}_2$ and ${\bar{X}}_3$ . ${\bar{X}}_1$ and ${\bar{X}}_3$ are in the tails of the hypothesized distribution of H ₀ . ${\stackrel{-}{X}}_2$ is within the (1-α) region where alpha is set by the analyst as the desired level of confidence in the test. Notice that the horizontal axis in the top panel is labeled $\bar{X}$ 's. This is the same theoretical distribution of $\bar{X}$ 's, the sampling distribution, that the Central Limit Theorem tells us is normally distributed. This is why we can draw it with this shape. The horizontal axis of the bottom panel is labeled Z and is the standard normal distribution. $Z_{\frac{\alpha }{2}}$ and ${\mathrm{-Z}}_{\frac{\alpha }{2}}$ , called the critical values , are marked on the bottom panel as the Z values associated with the probability the analyst has set as the level of confidence in the test, (1-α). The probabilities in the tails of both panels are, therefore, the same.

Notice that for each $\bar{X}$ there is an associated Z _c , called the calculated Z, that comes from solving the equation above. This calculated Z is nothing more than the number of standard deviations that the hypothesized mean is from the sample mean. If the sample mean falls "too many" standard deviations from the hypothesized mean we conclude that the sample mean could not have come from the distribution with the hypothesized mean, given our pre-set required level of confidence. It could have come from H ₀ , but it is deemed just too unlikely. If in fact this sample mean it did come from H ₀ , but from in the tail, we have made a Type I error. Our only real comfort is that we know the probability of making such an error, α, and we can control the size of α.