Question

the annual salaries (in dollars) of 14 randomly chosen fire fighters are listed. at α = 0.05, is there enough evidence support the claim that the standard deviation of the annual salaries is different from $5350? assume the population normally distributed. complete parts (a) through (e) below.
50,762 40,962 52,386 46,539 41,734 40,173 51,138
52,013 43,827 34,908 35,096 28,222 32,693 37,832
click the icon to view the chi - square distribution table.
(a) write the claim mathematically and identify h₀ and hₐ.
a. h₀: σ < 5350 (claim); hₐ: σ ≥ 5350
b. h₀: σ ≤ 5350 (claim); hₐ: σ > 5350
c. h₀: σ ≥ 5350; hₐ: σ < 5350 (claim)
d. h₀: σ = 5350; hₐ: σ ≠ 5350 (claim)
(b) find the critical value(s).
χ₀² =
(round to three decimal places as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Determine the degrees of freedom

The sample size \( n = 14 \), so the degrees of freedom \( df=n - 1=14 - 1 = 13\).

Step2: Determine the significance level for each tail

The significance level \( \alpha=0.05 \), and since it's a two - tailed test (because the alternative hypothesis is \( H_{a}:\sigma
eq5350\)), we split the significance level between the two tails. So \( \alpha/2=0.05/2 = 0.025\) and \( 1-\alpha/2 = 1 - 0.025=0.975\).

Step3: Find the critical values from the chi - square distribution table

We need to find \( \chi_{1-\alpha/2}^{2}\) and \( \chi_{\alpha/2}^{2}\) with \( df = 13\).

• For \( \chi_{0.975}^{2}\) with \( df = 13\), looking at the chi - square distribution table, we find that \( \chi_{0.975}^{2}=5.892\).
• For \( \chi_{0.025}^{2}\) with \( df = 13\), looking at the chi - square distribution table, we find that \( \chi_{0.025}^{2}=24.736\).

Answer:

(b) The critical values are $5.892, 24.736$