Question6.2.6. A random sample of size 16 is drawn from a normal distribution having $\sigma=6.0$ for the purpose of testing $H_{0}: \mu=30$ versus $H_{1}: \mu \neq 30$. The experimenter chooses to define the critical region $C$ to be the set of sample means lying in the interval (29.9, 30.1). What level of significance does the test have? Why is $(29.9,30.1)$ a poor choice for the critical region? What range of $\bar{y}$ values should comprise $C$, assuming the same $\alpha$ is to be used?

Studdy Solution
Determine the appropriate critical region for the same $\alpha$:
To find the appropriate critical region, we need to determine the z-scores that correspond to a total $\alpha$ of 0.0536 in the tails of the standard normal distribution. This requires finding the z-scores such that:
$P(Z < -z) + P(Z > z) = 0.0536$
This implies:
$P(Z > z) = 0.0268$
Using standard normal distribution tables or a calculator, find $z \approx 1.96$.
Thus, the appropriate critical region for the sample mean is:
$\bar{y} = \mu \pm z \cdot \sigma_{\bar{y}} = 30 \pm 1.96 \times 1.5$
$C = (27.06, 32.94)$
The appropriate critical region for the same $\alpha$ is $(27.06, 32.94)$.