Question

quarters are currently minted with weights normally distributed and having a standard deviation of 0.069. new equipment is being tested in an attempt to improve quality by reducing variation. a simple random sample of 30 quarters is obtained from those manufactured with the new equipment, and this sample has a standard deviation of 0.047. use a 0.05 significance level to test the claim that quarters manufactured with the new equipment have weights with a standard deviation less than 0.069. does the new equipment appear to be effective in reducing the variation of weights?

(b) find the critical value(s).
$chi_0^2 = \square$
(use a comma to separate answers as needed. round to three decimal places as needed.)
identify the rejection region(s).
\\(\bigcirc\\) a.
\\(\bigcirc\\) b.
\\(\bigcirc\\) c.
(images of distributions with shaded regions)
(c) use the \\(\chi^2\\)-test to find the standardized test statistic.

Explanation:

Step1: Determine test type and parameters

This is a left - tailed chi - square test for variance. The sample size \(n = 30\), so the degrees of freedom \(df=n - 1=29\). The significance level \(\alpha = 0.05\).

Step2: Find the critical value

For a left - tailed chi - square test, we need to find \(\chi_{1-\alpha,df}^{2}\). We want to find \(\chi_{0.95,29}^{2}\) (since \(\alpha = 0.05\) and it's left - tailed, the cumulative probability is \(1-\alpha=0.95\)). Using a chi - square distribution table or a calculator with chi - square inverse function, \(\chi_{0.95,29}^{2}\approx17.708\) (we can use the formula for chi - square critical values or refer to statistical software/ tables. For example, in a chi - square table, looking at \(df = 29\) and the column for \(0.95\) cumulative probability, we get the value).

Step3: Identify the rejection region

In a left - tailed chi - square test, the rejection region is the set of all test statistics \(\chi^{2}\) such that \(\chi^{2}<\chi_{1 - \alpha,df}^{2}\). Looking at the graphs, the graph with the left - tailed shaded region (the one where the shaded area is on the left side of the distribution) corresponds to the left - tailed test. So the rejection region is \(\chi^{2}<17.708\), and the correct graph for the rejection region is the one with the left - tailed shade (Graph B in the given options, assuming the middle graph is the left - tailed one as per the visual description where the shaded area is on the left).

Answer:

The critical value \(\chi_{0}^{2}=17.708\). The rejection region is \(\chi^{2}<17.708\) and the rejection region graph is the left - tailed one (Graph B).