Question

use a 0.025 significance level to test the claim that peanut candies have weights that vary more than plain candies. the standard deviation for the weights of plain candies is 0.331. a sample of 41 peanut candies has weights with a standard deviation of 0.28. assume the population is normally distributed.
(a) write the claim mathematically and identify h₀ and hₐ.
a. h₀: σ≤0.331; hₐ: σ>0.331 (claim)
b. h₀: σ≥0.331 (claim); hₐ: σ<0.331
c. h₀: σ=0.331; hₐ: σ≠0.331 (claim)
(b) find the critical value(s).
(use a comma to separate answers as needed. round to three decimal places as needed.)

Explanation:

Step1: Identify the test and parameters

We are testing a claim about the population standard deviation, so we use the chi - square test for variance (since standard deviation is related to variance). The sample size \(n = 41\), so the degrees of freedom \(df=n - 1=41-1 = 40\). The significance level \(\alpha=0.025\), and since the alternative hypothesis is \(H_{a}:\sigma>0.331\) (right - tailed test), we need to find the critical value \(\chi^{2}_{\alpha,df}\).

Step2: Find the critical value

We look up the chi - square distribution table for \(df = 40\) and \(\alpha=0.025\). Using a chi - square table or a statistical software, the critical value \(\chi^{2}_{0.025,40}\) is approximately 60.481.

Answer:

60.481