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 q is obtained from those manufactured with the new equipment, and this sample has a standard deviation of 0.044. 0.05 significance level to test the claim that quarters manufactured with the new equipment have weights with a st deviation less than 0.066. does the new equipment appear to be effective in reducing the variation of weights? (a) write the claim mathematically and identify h₀ and hₐ. a. h₀: σ≥0.066 (claim); hₐ: σ<0.066 b. h₀: σ≤0.066; hₐ: σ>0.066 (claim) c. h₀: σ≥0.066; hₐ: σ<0.066 (claim) d. h₀: σ≤0.066 (claim); hₐ: σ>0.066 (b) find the critical value(s). χ₀² = (use a comma to separate answers as needed. round to three decimal places as needed.)

Explanation:

Part (a)

The claim is that the standard deviation (\(\sigma\)) of weights with new equipment is less than \(0.066\), so the alternative hypothesis (\(H_a\)) is \(\sigma < 0.066\) (this is the claim). The null hypothesis (\(H_0\)) is the complement or the opposite for testing, so \(H_0: \sigma \geq 0.066\). Option C matches this setup.

Part (b)

Step1: Identify distribution and parameters

We are testing a hypothesis about a population standard deviation (or variance) with a sample size \(n = 28\), so the degrees of freedom \(df=n - 1=28 - 1 = 27\). The significance level \(\alpha = 0.05\), and the test is left - tailed (since \(H_a:\sigma<0.066\)). For a left - tailed test of variance/standard deviation, we use the chi - square distribution and find the critical value \(\chi_{1-\alpha,df}^2\).

Step2: Find the critical value

We need to find \(\chi_{0.95,27}^2\) (because \(\alpha = 0.05\) and it's a left - tailed test, so we use the \(1-\alpha = 0.95\) percentile). Using a chi - square table or a calculator with chi - square distribution functions, for \(df = 27\) and the 95th percentile (left - tailed critical value), we find that \(\chi_{0.95,27}^2=16.151\) (using chi - square distribution tables or statistical software: the chi - square distribution is right - skewed, and for \(df = 27\), the value corresponding to the lower 5% (since \(\alpha=0.05\) left - tailed) is approximately \(16.151\)).

Answer:

C. \( H_0: \sigma \geq 0.066 \); \( H_a: \sigma < 0.066 \) (Claim)