10. Hypothesis Testing for Two Samples

Two Variances - Graphing Calculator

10. Hypothesis Testing for Two Samples

Two Variances - Graphing Calculator

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Multiple Choice

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.
$H sub 0$ : _________ $H_{a}$ : ___ $P$ -value:
Because $P$ -value [ < | > ] $α$ , we [ REJECT | FAIL TO REJECT ] $H sub 0$ , there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

Because $P$ -value > $α$ , we FAIL TO REJECT $H sub 0$ , there is ENOUGH evidence to suggest $sigma sub 1 equals sigma sub 2$ .

Because $P$ -value > $α$ , we FAIL TO REJECT $H sub 0$ , there is NOT ENOUGH evidence to suggest $sigma sub 1 equals sigma sub 2$ .

Because $P$ -value < $α$ , we REJECT $H sub 0$ , there is ENOUGH evidence to suggest $sigma sub 1 is greater than sigma sub 2$ .

Because $P$ -value < $α$ , we REJECT $H sub 0$ , there is NOT ENOUGH evidence to suggest $sigma sub 1 is greater than sigma sub 2$ .

Verified step by step guidance

Identify the populations and the parameter of interest: We are comparing the variances (or standard deviations) of weights of rare coins from two independent samples: coins before 1900 and coins after 1900. The parameter of interest is the population variances $ \sigma_1^2 $ and $ \sigma_2^2 $.

State the null and alternative hypotheses based on the claim: Since the claim is that the variation before 1900 is greater than after 1900, the hypotheses are: \[ H_0: \sigma_1^2 \leq \sigma_2^2 \] \[ H_a: \sigma_1^2 > \sigma_2^2 \] where $ \sigma_1^2 $ is the variance before 1900 and $ \sigma_2^2 $ is the variance after 1900.

Choose the appropriate test and test statistic: Because we are comparing variances from two independent samples, use the F-test for equality of variances. The test statistic is: \[ F = \frac{s_1^2}{s_2^2} \] where $ s_1^2 $ and $ s_2^2 $ are the sample variances from the two groups. Make sure $ s_1^2 $ corresponds to the group hypothesized to have the larger variance (before 1900).

Determine the degrees of freedom for the numerator and denominator: \[ df_1 = n_1 - 1 \] \[ df_2 = n_2 - 1 \] where $ n_1 $ and $ n_2 $ are the sample sizes for the two groups respectively.

Use a graphing calculator or statistical software to find the p-value associated with the calculated F statistic and degrees of freedom. Compare the p-value to the significance level $ \alpha = 0.05 $: - If $ p \text{-value} < \alpha $, reject $ H_0 $ and conclude there is enough evidence to support that the variance before 1900 is greater. - If $ p \text{-value} \geq \alpha $, fail to reject $ H_0 $ and conclude there is not enough evidence to support the claim.

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