Q: Given the regression equation $$Y = 12.80 + 2.87X_1 + 41.25X_2$$, what is the predicted value of $$Y$$ when $$X_1 = 3$ and X_2$ = 2$$?

$$Y = 12.80 + 2.87 \times 3 + 41.25 \times 2 = 100.91$$

Question 1

Given the paired sample data for Company A and Company B, what is the correct null hypothesis ($$H_0) $$for testing if the mean difference is positive?

Accepted Answer

$$H_0$$: \mu_D = 0$$

Question 2

In a paired t-test, which formula is used to calculate the test statistic $$t_{stat}$$?

Accepted Answer

$$t_{stat} = \frac{\bar{D} - \mu_0$}{s_D / \sqrt{n}}$$

Question 3

A paired t-test is performed with $$n = 6$$ pairs, $$\bar{D} = 0.48$$, $$s_D = 0.40$$, and $$\mu_0$ = 0. $$What is the value of $$t_{stat}$$?

Accepted Answer

$$t_{stat} = \frac{0.48 - 0}{0.40 / \sqrt{6}} = 2.94$$

Question 4

If the critical value for a one-tailed t-test at $$\alpha = 0.05$$ and $$df = 5 is$$ $$2.015$$, and your calculated $$t_{stat} = 2.94$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$ because $$t_{stat} \u003e t_{critical}$$

Question 5

In the context of paired observations, what does $$\bar{D}$$ represent?

Accepted Answer

The mean of the differences between paired observations

Question 6

A human resources department wants to test if a new training program reduces absenteeism. They collect paired data before and after training. Which statistical test should they use?

Accepted Answer

Paired t-test

Question 7

Given the regression equation $$Y = 12.80 + 2.87X_1 + 41.25X_2$$, what is the predicted value of $$Y$$ when $$X_1 = 3$ and X_2$ = 2$$?

Accepted Answer

$$Y = 12.80 + 2.87 	imes 3 + 41.25 	imes 2 = 100.91$$

Question 8

What is the 90% confidence interval for the slope of $$X_1 if $$the interval is given as $$(0.51, 5.46)$$?

Accepted Answer

We are 90% confident that the true slope for $$X_1 is $$between $$0.51$$ and $$5.46$$

Question 9

If the 90% confidence interval for the mean response is $$(29.10, 55.30)$$, what does this interval represent?

Accepted Answer

We are 90% confident that the true mean response lies between $$29.10$$ and $$55.30$$

Question 10

Which of the following is NOT an assumption for the paired t-test?

Accepted Answer

The two samples are independent

Question 11

If the calculated $$t_{stat} is $$less than the negative critical value in a two-tailed test, what is the correct decision?

Accepted Answer

Reject $$H_0$$

Question 12

Which formula is used to calculate the confidence interval for the mean difference in a paired t-test?

Accepted Answer

$$\bar{D} \pm t_{\alpha/2, n-1} \cdot \frac{s_D}{\sqrt{n}}$$

Question 13

A regression output gives a coefficient for $$X_2 as$$ $$41.25$$ with a 90% confidence interval of $$(21.10, 55.30). $$What does this interval mean?

Accepted Answer

We are 90% confident that the true coefficient for $$X_2 is $$between $$21.10$$ and $$55.30$$

Question 14

If the p-value for a paired t-test is less than $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$

Question 15

Which of the following best describes the purpose of using paired observations in statistical analysis?

Accepted Answer

To control for variability between subjects by comparing each subject to themselves

Step-by-Step Guidance for Paired Observations and Regression in Statistics

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