Concerns about climate change and CO2 reduction have initiated the commercial production of blends of biodiesel (e.g., from renewable sources) and petrodiesel (from fossil fuel). Random samples of 48 blended fuels are tested in a lab to ascertain the bio/total carbon ratio.

(a) If the true mean is 0.9200 with a standard deviation of 0.0050, within what interval will 99 percent of the sample means fall?

Note: Round your answers to 4 decimal places.

To determine the interval within which 99% of the sample means will fall, we can use the concept of the sampling distribution of the sample mean. Specifically, we use the Central Limit Theorem and the formula for the confidence interval:

### Formula:
\[
\text{Confidence Interval} = \mu \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
\]

Where:
- \(\mu\) is the true mean.
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
- \(z_{\alpha/2}\) is the z-score corresponding to the desired confidence level (99% in this case).

### Step 1: Identify the given values
- True mean (\(\mu\)) = 0.9200
- Population standard deviation (\(\sigma\)) = 0.0050
- Sample size (\(n\)) = 48
- Confidence level = 99%

### Step 2: Find the z-score for a 99% confidence interval
For a 99% confidence interval, the z-score (\(z_{\alpha/2}\)) corresponds to 0.5% in each tail of the standard normal distribution. The z-score for 99% confidence is approximately 2.576.

### Step 3: Calculate the standard error (SE)
\[
\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}} = \frac{0.0050}{\sqrt{48}}
\]

### Step 4: Calculate the margin of error
\[
\text{Margin of Error} = z_{\alpha/2} \times \text{SE} = 2.576 \times \frac{0.0050}{\sqrt{48}}
\]

### Step 5: Calculate the confidence interval
\[
\text{Lower Bound} = \mu - \text{Margin of Error}
\]
\[
\text{Upper Bound} = \mu + \text{Margin of Error}
\]

Now, let's compute these values.It seems like I can’t do more advanced data analysis right now. Please try again later. 

Would you like to proceed with the calculations manually or do you have another question? 

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Here are 8 related questions you might find helpful:
1. How do you calculate the standard error of the mean for a sample?
2. What is the significance of the z-score in confidence intervals?
3. How does increasing the sample size affect the confidence interval?
4. Can you explain the Central Limit Theorem and its importance in statistics?
5. What are the differences between a confidence interval and a prediction interval?
6. How do you determine the z-score for different confidence levels?
7. What assumptions are necessary for constructing a confidence interval for the mean?
8. How would you interpret a 99% confidence interval in practical terms?

**Tip:** When calculating confidence intervals, ensure you have the correct z-score for your desired confidence level, as this directly affects the interval's width.

Calculate the 99% confidence interval for the bio/total carbon ratio based on a sample of 48 blended fuels. Understand the application of the Central Limit Theorem and the formula for confidence intervals in statistics. Useful for researchers and students studying statistical inference.

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