A historian is comparing the variation in weights of rare coins from different time periods. The data from two independent random samples is shown below. Using a 0.05 significance level and a graphing calculator, test the claim that the variation of weights before the 1900s is greater than after the 1900s.
$H_0$: _________ $H_a$: _________ $P$-value: ______
Because $P$-value [ < | > ] $\alpha$, we [ REJECT | FAIL TO REJECT ] $H_0$, there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

• A. Because $P$-value > $α$, we FAIL TO REJECT $H_0$, there is ENOUGH evidence to suggest $\sigma_1=\sigma_2$.
• B. Because $P$-value > $α$, we FAIL TO REJECT $H_0$, there is NOT ENOUGH evidence to suggest $\sigma_1=\sigma_2$.
• C. Because $P$-value < $α$, we REJECT $H_0$, there is ENOUGH evidence to suggest $\sigma_1>\sigma_2$.
• D. Because $P$-value < $α$, we REJECT $H_0$, there is NOT ENOUGH evidence to suggest $\sigma_1>\sigma_2$.

Answer: (C)

Verified step by step guidance

1. Step 1: Define the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)) for the test of two variances. Typically, \(H_0: \sigma_1^2 = \sigma_2^2\) (the variances are equal) and \(H_a\) could be \(\sigma_1^2 \neq \sigma_2^2\), \(\sigma_1^2 > \sigma_2^2\), or \(\sigma_1^2 < \sigma_2^2\) depending on the context.
2. Step 2: Collect the sample variances (\(s_1^2\) and \(s_2^2\)) and sample sizes (\(n_1\) and \(n_2\)) from the two independent samples.
3. Step 3: Calculate the test statistic using the formula for the F-test for equality of variances: \[F = \frac{s_1^2}{s_2^2}\] where \(s_1^2\) is the larger sample variance to ensure \(F \geq 1\).
4. Step 4: Determine the degrees of freedom for the numerator and denominator: \[df_1 = n_1 - 1\] \[df_2 = n_2 - 1\]
5. Step 5: Use the TI-84 calculator's built-in F-distribution functions to find the p-value corresponding to the calculated \(F\) statistic and degrees of freedom, then compare the p-value to the significance level (\(\alpha\)) to decide whether to reject or fail to reject the null hypothesis.