Two random samples were selected independently from populations having normal distributions. The statistics given below were extracted from the samples. Complete parts a through c. x₂ = 30.2 X₁ = 38.8 GOOOD a. If o₁ = 5 and 6₂ = 3 and the sample sizes are n₁ = 40 and n₂ = 40, construct a 99% confidence interval for the difference between the two population means. The confidence interval is (H₁-H₂) ≤ S (Round to two decimal places as needed.)
Two random samples were selected independently from populations having normal distributions. The statistics given below were extracted from the samples. Complete parts a through c. x₂ = 30.2 X₁ = 38.8 GOOOD a. If o₁ = 5 and 6₂ = 3 and the sample sizes are n₁ = 40 and n₂ = 40, construct a 99% confidence interval for the difference between the two population means. The confidence interval is (H₁-H₂) ≤ S (Round to two decimal places as needed.)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Constructing a Confidence Interval for the Difference Between Two Population Means**
In this exercise, we will learn how to construct a confidence interval for the difference between two population means based on sample data.
**Given Data:**
Two random samples were selected independently from populations having normal distributions. The statistics given below were extracted from the samples.
- Sample Mean 1 (\(\bar{x}_1\)) = 38.8
- Sample Mean 2 (\(\bar{x}_2\)) = 30.2
**Question:**
a. Given the population standard deviations (\(\sigma_1 = 5\) and \(\sigma_2 = 3\)) and the sample sizes (\(n_1 = 40\) and \(n_2 = 40\)), construct a 99% confidence interval for the difference between the two population means.
**Steps to Calculate the Confidence Interval:**
1. **Identify the Confidence Level:**
- The confidence level is 99%.
2. **Calculate the Standard Error (SE) for the Difference of Means:**
\[
SE = \sqrt{\left(\frac{\sigma_1^2}{n_1}\right) + \left(\frac{\sigma_2^2}{n_2}\right)}
\]
Substituting the given values:
\[
SE = \sqrt{\left(\frac{5^2}{40}\right) + \left(\frac{3^2}{40}\right)} = \sqrt{\left(\frac{25}{40}\right) + \left(\frac{9}{40}\right)} = \sqrt{\frac{34}{40}} = \sqrt{0.85}
\]
\[
SE \approx 0.92
\]
3. **Identify the Z-Score for the 99% Confidence Level:**
- For a 99% confidence level, the corresponding Z-score is approximately 2.576.
4. **Calculate the Margin of Error (ME):**
\[
ME = Z \times SE = 2.576 \times 0.92 \approx 2.37
\]
5. **Determine the Confidence Interval:**
- The difference between the two sample means is \(\bar{x}_1 - \bar{x}_2 = 38.8 - 30.2 = 8.6\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbdd081ff-4988-4514-ab5d-41964c228e08%2F342b09a6-f017-4b98-b8c7-e988128d4744%2F1m3cv9r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Constructing a Confidence Interval for the Difference Between Two Population Means**
In this exercise, we will learn how to construct a confidence interval for the difference between two population means based on sample data.
**Given Data:**
Two random samples were selected independently from populations having normal distributions. The statistics given below were extracted from the samples.
- Sample Mean 1 (\(\bar{x}_1\)) = 38.8
- Sample Mean 2 (\(\bar{x}_2\)) = 30.2
**Question:**
a. Given the population standard deviations (\(\sigma_1 = 5\) and \(\sigma_2 = 3\)) and the sample sizes (\(n_1 = 40\) and \(n_2 = 40\)), construct a 99% confidence interval for the difference between the two population means.
**Steps to Calculate the Confidence Interval:**
1. **Identify the Confidence Level:**
- The confidence level is 99%.
2. **Calculate the Standard Error (SE) for the Difference of Means:**
\[
SE = \sqrt{\left(\frac{\sigma_1^2}{n_1}\right) + \left(\frac{\sigma_2^2}{n_2}\right)}
\]
Substituting the given values:
\[
SE = \sqrt{\left(\frac{5^2}{40}\right) + \left(\frac{3^2}{40}\right)} = \sqrt{\left(\frac{25}{40}\right) + \left(\frac{9}{40}\right)} = \sqrt{\frac{34}{40}} = \sqrt{0.85}
\]
\[
SE \approx 0.92
\]
3. **Identify the Z-Score for the 99% Confidence Level:**
- For a 99% confidence level, the corresponding Z-score is approximately 2.576.
4. **Calculate the Margin of Error (ME):**
\[
ME = Z \times SE = 2.576 \times 0.92 \approx 2.37
\]
5. **Determine the Confidence Interval:**
- The difference between the two sample means is \(\bar{x}_1 - \bar{x}_2 = 38.8 - 30.2 = 8.6\).
Transcribed Image Text:Consider the following sample information randomly selected from two populations.
| Sample 1 | Sample 2 |
|------------|------------|
| n₁ = 250 | n₂ = 100 |
| x₁ = 30 | x₂ = 33 |
**a.** Determine if the sample sizes are large enough so that the sampling distribution for the difference between the sample proportions is approximately normally distributed.
**b.** Calculate a 95% confidence interval for the difference between the two population proportions.
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