Question

quarters are currently minted with weights normally distributed and having a standard deviation of 0.066. new equipment is being tested in an attempt to improve quality by reducing variation. a simple random sample of 28 quarters is obtained from those manufactured with the new equipment, and this sample has a standard deviation of 0.044. 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 0.066. does the new equipment appear to be effective in reducing the variation of weights? (b) find the critical value(s). \\(\chi_0^2 = 16.151\\) (use a comma to separate answers as needed. round to three decimal places as needed.) identify the rejection region(s). a. image of a distribution with a shaded left tail b. image of a distribution with a shaded left area c. image of a distribution with two shaded tails (c) use the \\(\chi^2\\)-test to find the standardized test statistic. \\(\chi^2 = \square\\) (round to three decimal places as needed.)

Explanation:

Step1: Recall the formula for the chi - square test statistic for standard deviation

The formula for the chi - square test statistic \(\chi^{2}\) when testing a claim about the standard deviation (or variance) of a normally distributed population is \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma^{2}}\), where \(n\) is the sample size, \(s\) is the sample standard deviation, and \(\sigma\) is the population standard deviation.

Step2: Identify the values of \(n\), \(s\), and \(\sigma\)

We are given that \(n = 28\) (the sample size), \(s=0.044\) (the sample standard deviation), and \(\sigma = 0.066\) (the population standard deviation).

Step3: Calculate \((n - 1)\), \(s^{2}\), and \(\sigma^{2}\)

First, calculate \(n-1\): \(n - 1=28 - 1=27\).
Then, calculate \(s^{2}\): \(s^{2}=(0.044)^{2}=0.001936\).
Next, calculate \(\sigma^{2}\): \(\sigma^{2}=(0.066)^{2}=0.004356\).

Step4: Substitute the values into the formula

Substitute \(n - 1 = 27\), \(s^{2}=0.001936\), and \(\sigma^{2}=0.004356\) into the formula \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma^{2}}\):

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Answer:

11.459