https://file.aiquickdraw.com/mathbot/file/1730258948003yfz0n9q9.jpg

The problem requires calculating the value of the test statistic \( t \) for a hypothesis test.

### Given Information:
- Null Hypothesis: \( H_0: \mu = 20 \)
- Alternative Hypothesis: \( H_1: \mu > 20 \)
- Sample Size (\( n \)) = 39
- Sample Mean (\( \bar{x} \)) = 24
- Sample Standard Deviation (\( s \)) = 2

### Formula for the t-test statistic:
The t-test statistic is calculated as:
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\]
where:
- \( \bar{x} \) is the sample mean,
- \( \mu \) is the population mean under the null hypothesis,
- \( s \) is the sample standard deviation, and
- \( n \) is the sample size.

### Substituting the values:
1. Difference between the sample mean and the hypothesized mean:
   \[
   \bar{x} - \mu = 24 - 20 = 4
   \]
2. Standard error of the mean:
   \[
   \frac{s}{\sqrt{n}} = \frac{2}{\sqrt{39}} \approx \frac{2}{6.244} \approx 0.32
   \]
3. Calculating \( t \):
   \[
   t = \frac{4}{0.32} \approx 12.50
   \]

### Answer:
The value of the test statistic \( t \) is approximately \( 12.50 \).

Would you like further details or explanations on hypothesis testing or related topics?

#### Related Questions:
1. How do we interpret the results of a t-test in hypothesis testing?
2. What is the significance of the degrees of freedom in a t-test?
3. How do we determine the critical t-value for this test?
4. What are Type I and Type II errors in hypothesis testing?
5. How does sample size impact the test statistic and the conclusion of a hypothesis test?

#### Tip:
Always ensure your sample size is large enough to approximate the normal distribution in t-tests, which helps in obtaining reliable results.

A test is made of H₀: μ = 20 versus H₁: μ > 20. A sample of size 39 is drawn. The sample mean and standard deviation are x̄ = 24 and s = 2. Compute the value of the test statistic t. Round your answer to two decimal places.

Learn how to compute the t-statistic for a hypothesis test where H₀: μ = 20 and H₁: μ > 20, using a sample of size 39 with a mean of 24 and standard deviation of 2. This guide covers essential statistical concepts, including hypothesis testing and t-statistics, ideal for undergraduate statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation