valuate the following formula for x₁ = 28.4215, x₂ = 26.0486, H₁-H2=0, Sp = 40.93, n₁ = 50, and n₂ = 47. (x₁-x₂)-(H1-H₂) 2 Р n₁ + 2 Р n₂ (Round to two decimal places as needed.) ...
valuate the following formula for x₁ = 28.4215, x₂ = 26.0486, H₁-H2=0, Sp = 40.93, n₁ = 50, and n₂ = 47. (x₁-x₂)-(H1-H₂) 2 Р n₁ + 2 Р n₂ (Round to two decimal places as needed.) ...
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 28E
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![## Example T-Test Calculation:
Evaluate the following formula with the given values:
- \( x_1 = 28.4215 \)
- \( x_2 = 26.0486 \)
- \( \mu_1 - \mu_2 = 0 \)
- \( s_p = 40.93 \)
- \( n_1 = 50 \)
- \( n_2 = 47 \)
The formula to evaluate is:
\[
t = \frac{(x_1 - x_2) - (\mu_1 - \mu_2)}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}
\]
### Explanation:
In this formula:
- \( x_1 \) and \( x_2 \) are sample means.
- \( \mu_1 \) and \( \mu_2 \) are population means.
- \( s_p \) is the pooled standard deviation.
- \( n_1 \) and \( n_2 \) are the sample sizes.
### Calculation Steps:
1. Subtract \( x_2 \) from \( x_1 \):
\[ x_1 - x_2 = 28.4215 - 26.0486 = 2.3729 \]
2. Subtract the population mean difference from the result:
\[ (x_1 - x_2) - (\mu_1 - \mu_2) = 2.3729 - 0 = 2.3729 \]
3. Calculate the squared term for the sample sizes:
\[ \frac{1}{n_1} + \frac{1}{n_2} = \frac{1}{50} + \frac{1}{47} \approx 0.0206 + 0.0213 = 0.0419 \]
4. Calculate the square root of the sum:
\[ \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} = \sqrt{0.0419} \approx 0.2047 \]
5. Multiply the pooled standard deviation by the result:
\[ s_p \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} = 40.93 \times 0.2047 \approx 8.38 \]
6](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39618134-6ceb-48ac-ab5a-e0c16873beed%2F6ff6a865-9eec-4cfa-a576-39e10d8325fd%2Fpnsj99q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Example T-Test Calculation:
Evaluate the following formula with the given values:
- \( x_1 = 28.4215 \)
- \( x_2 = 26.0486 \)
- \( \mu_1 - \mu_2 = 0 \)
- \( s_p = 40.93 \)
- \( n_1 = 50 \)
- \( n_2 = 47 \)
The formula to evaluate is:
\[
t = \frac{(x_1 - x_2) - (\mu_1 - \mu_2)}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}
\]
### Explanation:
In this formula:
- \( x_1 \) and \( x_2 \) are sample means.
- \( \mu_1 \) and \( \mu_2 \) are population means.
- \( s_p \) is the pooled standard deviation.
- \( n_1 \) and \( n_2 \) are the sample sizes.
### Calculation Steps:
1. Subtract \( x_2 \) from \( x_1 \):
\[ x_1 - x_2 = 28.4215 - 26.0486 = 2.3729 \]
2. Subtract the population mean difference from the result:
\[ (x_1 - x_2) - (\mu_1 - \mu_2) = 2.3729 - 0 = 2.3729 \]
3. Calculate the squared term for the sample sizes:
\[ \frac{1}{n_1} + \frac{1}{n_2} = \frac{1}{50} + \frac{1}{47} \approx 0.0206 + 0.0213 = 0.0419 \]
4. Calculate the square root of the sum:
\[ \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} = \sqrt{0.0419} \approx 0.2047 \]
5. Multiply the pooled standard deviation by the result:
\[ s_p \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} = 40.93 \times 0.2047 \approx 8.38 \]
6
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