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. $h_0: \sigma \leq 0.331$; $h_a: \sigma > 0.331$ (claim) b. $h_0: \sigma \geq 0.331$ (claim); $h_a: \sigma < 0.331$ c. $h_0: \sigma = 0.331$; $h_a: \sigma \
eq 0.331$ (claim) (b) find the critical value(s). 59.342 (use a comma to separate answers as needed. round to three decimal places as needed.) identify the rejection region(s). a. b. c.

Explanation:

(a) Step1: Identify Hypotheses

The claim is that peanut candies have weights that vary more than plain candies. Variation is measured by standard deviation (\(\sigma\)). So, we test if \(\sigma\) (peanut) \(> \sigma\) (plain) = 0.331. Thus, \( H_0: \sigma \leq 0.331 \) (null, no more variation) and \( H_a: \sigma > 0.331 \) (claim, more variation). Option A matches.

(b) Step1: Determine Distribution

This is a right - tailed chi - square test for standard deviation. For a sample size \( n = 41 \), degrees of freedom \( df=n - 1=41 - 1 = 40 \). Significance level \(\alpha = 0.025\) (right - tailed).

(b) Step2: Find Critical Value

Using chi - square distribution table or calculator, for \( df = 40 \) and \(\alpha=0.025\) (right - tail), the critical value \(\chi^2_{0.025,40}=59.342\).

(b) Step3: Identify Rejection Region

In a right - tailed chi - square test, the rejection region is the area to the right of the critical value. Among the three graphs, Graph B (the middle one) has the rejection region (shaded area) to the right of the critical value, which matches a right - tailed test.

Answer:

(a)

A. \( H_0: \sigma \leq 0.331 \); \( H_a: \sigma > 0.331 \) (Claim)

(b)

Critical value: \( 59.342 \)
Rejection region: The Middle Graph (Graph B)