10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

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Multiple Choice

Create a confidence interval with $alpha equals 0.1$ for the difference between the 2 population means to see if there's evidence that $mu sub 1 is not equal to mu sub 2$ .

Confidence interval = $\left(-4.02,5.06\right)$ ; includes $0$ , not enough evidence to suggest $\mu_1\ne\mu_2$ .

Confidence interval = $\left(-4.02,5.06\right)$ ; enough evidence to suggest $\mu_1\ne\mu_2$ .

Confidence interval = $\left(4.02,5.06\right)$ ; not enough evidence to suggest $\mu_1\ne\mu_2$ .

Confidence interval = $\left(4.02,5.06\right)$ ; enough evidence to suggest $\mu_1\ne\mu_2$ .

Verified step by step guidance

Calculate the sample means ($ \bar{x}_1 $ and $ \bar{x}_2 $) for Sample 1 and Sample 2 by summing the values in each sample and dividing by the number of observations (7 in each sample).

Calculate the sample standard deviations ($ s_1 $ and $ s_2 $) for each sample using the formula for sample standard deviation: $ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} $, where $ n $ is the sample size.

Determine the standard error (SE) of the difference between the two sample means using the formula: \[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \], where $ n_1 $ and $ n_2 $ are the sample sizes.

Find the critical t-value ($ t_{\alpha/2} $) for a two-tailed test with $ \alpha = 0.1 $ and degrees of freedom approximated using the smaller of $ n_1 - 1 $ and $ n_2 - 1 $ or using the Welch-Satterthwaite equation for unequal variances.

Construct the confidence interval for the difference between the population means using the formula: \[ (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \times SE \]. This interval will help determine if zero is included, indicating whether there is evidence that $ \mu_1 \neq \mu_2 $.