Perform a hypothesis test with α=0.1\alpha=0.1 to see if there is evidence that μ1≠μ2\mu_1\ne\mu_2

P − v a l u e > α P-value>\alpha P − v a l u e > α ; fail to reject H 0 H_0 H 0 since there is not enough evidence to suggest μ 1 ≠ μ 2 \mu_1\ne\mu_2 μ 1  = μ 2 .

Perform a hypothesis test with α = 0.1 \alpha=0.1 to see if there is evidence that μ 1 ≠ μ 2 \mu_1\ne\mu_2

Welcome back. For this problem, we'd like to perform a hypothesis test with alpha equals point one to see if there's evidence that mu one is not equal to mu two. So let's start by writing our hypotheses. The null hypothesis is going to be that the two means are equal, where mu one is equal to mu two. And the alternative hypothesis is what we're trying to see if there's enough evidence to suggest, which in this case is that mu one is not equal to mu two. Now that we have our hypothesis, let's perform the hypothesis test. We will need to start with step one because this time we have been given data. So we need to enter it in by clicking on the stat button, clicking on enter, and then putting in the data. You wanna put sample one data in l one and sample two data in l two. As you can see, I've already done that, but you should pause the video here and put the data in for yourself. Once that's all set, that's step one done, and we're ready to move to step two. Clicking on the stat button, scrolling two to the right to the test menu, and then clicking on four. That's step two. All set. We'll follow the instructions on the left because we've been given data, so make sure the flashing cursor is over data and click enter. Then you wanna make sure that list one is where you put sample one data, l one, and list two is where you put sample two data, l two. Freak one and freak two should both be set to one. Then you wanna make sure that the correct alternative hypothesis has been chosen. In this case, it's that mu one is not equal to mu two, so make sure that not equal to mu two has been selected. No should be selected for pooled, and you can scroll down to calculate and click enter. That's step three. All set. We end up getting a p value of approximately 0.75, which we wanna compare to 0.1, our alpha value. We can see that our p value is larger than alpha, so we fail to reject the null hypothesis, meaning that there is not enough evidence to suggest that the alternative hypothesis is true, which in this case is that mu one is not equal to mu two. Great work.

Perform a hypothesis test with α = 0.1 \alpha=0.1 to see if there is evidence that μ 1 ≠ μ 2 \mu_1\ne\mu_2

P − v a l u e > α P-value>\alpha P − v a l u e > α ; fail to reject H 0 H_0 H 0 since there is not enough evidence to suggest μ 1 ≠ μ 2 \mu_1\ne\mu_2 μ 1  = μ 2 .

P − v a l u e < α P-value<\alpha P − v a l u e < α ; fail to reject H 0 H_0 H 0 since there is not enough evidence to suggest μ 1 ≠ μ 2 \mu_1\ne\mu_2 μ 1  = μ 2 .

P − v a l u e < α P-value<\alpha P − v a l u e < α ; reject H 0 H_0 H 0 since there is enough evidence to suggest μ 1 ≠ μ 2 \mu_1\ne\mu_2 μ 1  = μ 2 .

P − v a l u e > α P-value>\alpha P − v a l u e > α ; reject H 0 H_0 H 0 since there is enough evidence to suggest μ 1 ≠ μ 2 \mu_1\ne\mu_2 μ 1  = μ 2 .

10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

Statistics for Business

10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

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Multiple Choice

Perform a hypothesis test with $alpha equals 0.1$ to see if there is evidence that $mu sub 1 is not equal to mu sub 2$

$P-value<\alpha$ ; fail to reject $H_0$ since there is not enough evidence to suggest $\mu_1\ne\mu_2$ .

$P-value<\alpha$ ; reject $H_0$ since there is enough evidence to suggest $\mu_1\ne\mu_2$ .

$P-value>\alpha$ ; fail to reject $H_0$ since there is not enough evidence to suggest $\mu_1\ne\mu_2$ .

$P-value>\alpha$ ; reject $H_0$ since there is enough evidence to suggest $\mu_1\ne\mu_2$ .

Verified step by step guidance

Step 1: Define the hypotheses for the test. The null hypothesis (H₀) is that the population means are equal:

H

₀:μ₁=μ₂. The alternative hypothesis (H₁) is that the population means are not equal:

H

₁:μ₁≠μ₂.

Step 2: Calculate the sample means and sample standard deviations for both samples using the data provided. For each sample, compute the mean

\bar{x}

and standard deviation

s

Step 3: Determine the appropriate test statistic. Since the samples are independent and the population variances are unknown, use the two-sample t-test formula for the test statistic

t

= \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}

, where

n_1

and

n_2

are the sample sizes.

Step 4: Calculate the degrees of freedom for the test using the formula for unequal variances (Welch's t-test):

d.f. = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}}

Step 5: Using the calculated t-statistic and degrees of freedom, find the p-value from the t-distribution. Compare the p-value to the significance level

\alpha = 0.1

. If

p\text{-value} > \alpha

, fail to reject the null hypothesis; otherwise, reject the null hypothesis.

8:24m

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