Hypothesis testing
04/14/2021
Decision citeria for hypothesis testing
- In the last slide, we proposed the criterion that observing a sample mean \( 1.9 \) standard deviations away from the hypothesized center \( \mu=50 \) to be reason to question the hypothesis.
- This corresponded to a decision that if \( 48.5 \leq \overline{x} \leq 51.5, \) we will not reject the null hypothesis \( H_0 \), and if either \[ \overline{x} < 48.5 \text{ or }\overline{x} > 51.5, \] we will reject the null hypothesis in favor of the alternative hypothesis \( H_1 \).
- This is illustrated to the right.
Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition
- The values of \( \overline{x} \) that are less than \( 48.5 \) and greater than \( 51.5 \) constitute the critical region for the test;
- all values that are in the interval \[ 48.5 \leq \overline{x} \leq 51.5 \] form a region for which we will fail to reject the null hypothesis.
- By convention, this is usually called the acceptance region.
- The boundaries between the critical regions and the acceptance region are called the critical values – in our example, the critical values are \( 48.5 \) and \( 51.5 \).
- It is customary to state conclusions relative to the null hypothesis \( H_0 \).
- Therefore, we reject \( H_0 \) in favor of \( H_1 \) if the test statistic falls in the critical region and fails to reject \( H_0 \) otherwise.
Decision citeria for hypothesis testing – types of errors
- Our decision process is based on the random outcome of the test statistic, so that even if an outcome seems unlikely, we may come to a false conclusion based on observing a low-probability event.
- There are two possible wrong conclusions we can make in this decision process:
- we may reject the null hypothesis when this is actually true;
- we may fail to reject the null hypothesis when this is actually false.
- In our example, the true mean burning rate of the propellant could be equal to \( 50 \) centimeters per second;
- yet for the specimens that are tested, we could observe a value of the test statistic \( \overline{x} \) that falls into the critical region.
Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition
- We would then reject the null hypothesis \( H_0 \) in favor of the alternate \( H_1 \) when, in fact, \( H_0 \) is really true.
- This type of wrong conclusion is called a type I error.
Type I Error
Rejecting the null hypothesis \( H_0 \) when it is true is defined as a type I error.
Decision citeria for hypothesis testing – types of errors
- Now suppose that the true mean burning rate is different from 50 centimeters per second
- yet for the specimens that are tested, we could observe a value of the test statistic \( \overline{x} \) that falls into the acceptance region.
Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition
- We would then fail to reject the null hypothesis \( H_0 \) when, in fact, \( H_0 \) is really false.
- This type of wrong conclusion is called a type II error.
- A schematic of this hypothesis testing decision process is given in the right:
Type II Error
Failing to reject the null hypothesis \( H_0 \) when it is false is defined as a type II error.
Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition
- Based on these two possible errors, we can define different probabilistic criteria that will attempt to handle these risks of incorrect decisions.
- It turns out that the rate of failure of confidence intervals is the same as the probability of type I error.
Probability of Type I Error
\[ \alpha = P(\text{type I error}) = P(\text{reject }H_0\text{ when }H_0\text{ is true}) \]
Decision citeria for hypothesis testing – types of errors
- If we consider how we constructed the confidence interval, the duality makes sense once again.
- In our earlier example, we had \( H_0: \mu = 50 \) and \( H_1 : \mu \neq 50 \).
- We define our decision criterion in that we will reject the null hypothesis if \( \overline{x} \) lies outside of a stated interior region around the assumed \( \mu=50 \).
Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition
- This region is constructed by considering the distribution for the test statistic, i.e., let’s suppose for simplicity that \( \sigma \) is known and \[ \overline{X} \sim N\left(\mu, \sigma^2\right). \]
- Assuming \( \mu =50 \), we can compute critical values \( z_\frac{\alpha}{2} \) for which \[ \begin{align} P\left(\overline{X} < \overline{x}_l\right) = P\left(\overline{X} < 50 - \sigma_\overline{X} z_\frac{\alpha}{2}\right) & = \frac{\alpha}{2}\\ P\left(\overline{X} > \overline{x}_u\right) = P\left(\overline{X} > 50 - \sigma_\overline{X} z_\frac{\alpha}{2}\right) & = \frac{\alpha}{2}. \end{align} \]
- Recall that \( \sigma_\overline{X} z_\frac{\alpha}{2} \) is precisely the radius of the confidence interval.
- Therefore, the confidence interval centered at \( \overline{x} \) does not contain \( \mu=50 \) if and only if \[ \begin{align} \overline{x} < 50 - \sigma_\overline{X} z_\frac{\alpha}{2} && \text{ or } \overline{x} > 50 + \sigma_\overline{X} z_\frac{\alpha}{2} \end{align} \]
- I.e., we reject the null hypothesis if and only if the test statistic is observed beyond the critical values for the sampling distribution.