Based on all student records at Camford University, students spend an average of 6.00 hours per week playing organized sports. The population’s standard deviation is 3.40 hours per week. Based on a sample of 49 students, Healthy Lifestyles Incorporated (HLI) would like to apply the central limit theorem to make various estimates.

Calculate the probability that the sample mean will be between 5.6 and 6.4 hours.
Note: Round your z and standard error values to 2 decimal places. Round your intermediate and final answer to 4 decimal places.



Given:
- Population mean (\(\mu\)) = 6.00 hours
- Population standard deviation (\(\sigma\)) = 3.40 hours
- Sample size (\(n\)) = 49
- Sample mean range: 5.6 to 6.4 hours

We need to calculate the probability that the sample mean is between 5.6 and 6.4 hours. Let's break this into steps:

### Step 1: Find the standard error of the mean (SE)
The standard error of the mean is calculated as:

\[
SE = \frac{\sigma}{\sqrt{n}}
\]

### Step 2: Convert the sample means (5.6 and 6.4) to z-scores
The z-score for a sample mean (\(X\)) is given by:

\[
z = \frac{X - \mu}{SE}
\]

### Step 3: Find the probabilities corresponding to the z-scores
The probability can then be found using the standard normal distribution table.

Let's go through these calculations now.### Step-by-step Calculation:

1. **Standard Error (SE):** 

\[
SE = \frac{3.40}{\sqrt{49}} = 0.49 \quad \text{(rounded to 2 decimal places)}
\]

2. **Z-scores:**
   - For \(X_1 = 5.6\):

\[
z_1 = \frac{5.6 - 6.00}{0.49} \approx -0.82
\]

   - For \(X_2 = 6.4\):

\[
z_2 = \frac{6.4 - 6.00}{0.49} \approx 0.82
\]

3. **Probability:** 
   - The probability corresponding to \(z_1 = -0.82\) is approximately 0.2061.
   - The probability corresponding to \(z_2 = 0.82\) is approximately 0.7939.

4. **Probability that the sample mean is between 5.6 and 6.4:**

\[
P(5.6 < \bar{X} < 6.4) = 0.7939 - 0.2061 = 0.5898
\]

### Final Answer:

The probability that the sample mean will be between 5.6 and 6.4 hours is **0.5898** (rounded to 4 decimal places).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the probability that the sample mean will be less than 5.6 hours?
2. How does increasing the sample size to 100 affect the standard error?
3. What is the probability that the sample mean will be more than 6.4 hours?
4. If the sample size is decreased to 25, how does that change the probability calculation?
5. How would the probability change if the population's standard deviation was smaller?

**Tip:** The Central Limit Theorem helps us to estimate probabilities about the sample mean, especially when the sample size is large.

Based on all student records at Camford University, students spend an average of 6.00 hours per week playing organized sports. The population’s standard deviation is 3.40 hours per week. Based on a sample of 49 students, Healthy Lifestyles Incorporated (HLI) would like to apply the central limit theorem to make various estimates.

Calculate the probability that the sample mean will be between 5.6 and 6.4 hours.

Note: Round your z and standard error values to 2 decimal places. Round your intermediate and final answer to 4 decimal places.

Find out how to calculate the probability that the sample mean of sports activity hours for students at Camford University is between 5.6 and 6.4 using the Central Limit Theorem. The problem includes key statistical concepts such as the standard error, z-scores, and probability distribution. Suitable for college-level 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