Consider the following hypothesis test.
H0: 𝜇d ≤ 0
Ha: 𝜇d > 0
(a)
The following data are from matched samples taken from two populations. Compute the difference value for each element. (Use Population 1 − Population 2.)
Element	Population	Difference
1	2
1	21	19	
2	28	28	
3	18	18	
4	20	19	
5	26	24	
(b)
Compute 
d.
(c)
Compute the standard deviation 
sd.
(d)
Conduct a hypothesis test using 
𝛼 = 0.05.
Calculate the test statistic. (Round your answer to three decimal places.)
Calculate the p-value. (Round your answer to four decimal places.)
p-value = 


To tackle the given hypothesis test step-by-step, we need to follow these instructions:

### (a) Compute the difference value for each element
The differences are calculated as \( \text{Population 1} - \text{Population 2} \).

\[
\begin{array}{|c|c|c|c|}
\hline
\text{Element} & \text{Population 1} & \text{Population 2} & \text{Difference} \\
\hline
1 & 21 & 19 & 21 - 19 = 2 \\
2 & 28 & 28 & 28 - 28 = 0 \\
3 & 18 & 18 & 18 - 18 = 0 \\
4 & 20 & 19 & 20 - 19 = 1 \\
5 & 26 & 24 & 26 - 24 = 2 \\
\hline
\end{array}
\]

So, the differences are: \(2, 0, 0, 1, 2\).

### (b) Compute \(\bar{d}\)
The mean of the differences, \(\bar{d}\), is computed as:

\[
\bar{d} = \frac{\sum d_i}{n}
\]

\[
\bar{d} = \frac{2 + 0 + 0 + 1 + 2}{5} = \frac{5}{5} = 1
\]

### (c) Compute the standard deviation \( s_d \)
The standard deviation of the differences is computed as:

\[
s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}}
\]

First, we find the squared differences from the mean:

\[
\begin{array}{|c|c|c|}
\hline
\text{Difference} & \text{Difference from Mean} & \text{Squared Difference} \\
\hline
2 & 2 - 1 = 1 & 1^2 = 1 \\
0 & 0 - 1 = -1 & (-1)^2 = 1 \\
0 & 0 - 1 = -1 & (-1)^2 = 1 \\
1 & 1 - 1 = 0 & 0^2 = 0 \\
2 & 2 - 1 = 1 & 1^2 = 1 \\
\hline
\end{array}
\]

Sum of squared differences:

\[
\sum (d_i - \bar{d})^2 = 1 + 1 + 1 + 0 + 1 = 4
\]

Now compute \( s_d \):

\[
s_d = \sqrt{\frac{4}{5-1}} = \sqrt{1} = 1
\]

### (d) Conduct a hypothesis test using \( \alpha = 0.05 \)

**Calculate the test statistic**

The test statistic for the mean of differences is given by:

\[
t = \frac{\bar{d}}{s_d / \sqrt{n}}
\]

\[
t = \frac{1}{1 / \sqrt{5}} = \frac{1}{1 / 2.236} = 2.236
\]

**Calculate the p-value**

We need to find the p-value for the one-tailed test with 4 degrees of freedom. Using the t-distribution table or a t-distribution calculator, for \( t = 2.236 \) with 4 degrees of freedom:

The p-value = 0.0464

### Summary

- **Test Statistic (t)**: \( 2.236 \)
- **p-value**: \( 0.0464 \)

Since \( p = 0.0464 < \alpha = 0.05 \), we reject the null hypothesis \( H_0 \). There is sufficient evidence to conclude that the mean difference \( \mu_d \) is greater than 0.

---

Do you want more details on any part of the solution, or do you have any questions?

Here are 8 related questions to expand your understanding:

1. How is the standard deviation of differences \( s_d \) used in the hypothesis test?
2. What does it mean to reject the null hypothesis in this context?
3. How would the test statistic change if the standard deviation were larger?
4. Why do we use the t-distribution instead of the normal distribution in this test?
5. What would be the implication if \( \alpha \) were set to 0.01 instead of 0.05?
6. How do we interpret the p-value in the context of this hypothesis test?
7. What assumptions are made when conducting this paired t-test?
8. How might outliers in the data affect the results of this test?

**Tip:** Always check the assumptions of the hypothesis test before drawing conclusions.

Explore how to conduct hypothesis testing using paired samples and compute the test statistic and p-value. Includes step-by-step calculations and interpretation of results. Suitable for advanced college-level students and researchers.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation