Use the estimation method to calculate and interpret a confidence interval. (3 pts.) Suppose we are sampling from a population that is known to be normal with a standard deviation of o = 5. However, the mean is unknown, so we will need to estimate it using our sample mean. A. For a sample mean (X) = 20 and a sample size N = 25, compute a 95% confidence interval for (Show the process you used to determine the CI bounds.)

To compute a 95% confidence interval (CI) for the population mean given that the population is normally distributed with a known standard deviation \( \sigma = 5 \), a sample mean \( \bar{X} = 20 \), and a sample size \( N = 25 \), we'll follow these steps:

### Step-by-Step Process:

1. **Determine the Z-score** for a 95% confidence level:
   - For a 95% confidence interval, the corresponding Z-score (critical value) is found using standard normal distribution tables. The Z-value for 95% confidence is approximately \( Z = 1.96 \).
   
2. **Calculate the standard error of the mean (SE)**:
   \[
   SE = \frac{\sigma}{\sqrt{N}} = \frac{5}{\sqrt{25}} = \frac{5}{5} = 1
   \]
   
3. **Compute the margin of error (ME)**:
   \[
   ME = Z \times SE = 1.96 \times 1 = 1.96
   \]

4. **Calculate the confidence interval bounds**:
   The formula for the confidence interval is:
   \[
   \left( \bar{X} - ME, \bar{X} + ME \right)
   \]
   Substituting the values:
   \[
   \left( 20 - 1.96, 20 + 1.96 \right) = \left( 18.04, 21.96 \right)
   \]

### Conclusion:
The 95% confidence interval for the population mean is \( (18.04, 21.96) \). This means we are 95% confident that the true population mean lies within this interval.

### Interpretation:
If we were to take many samples of size 25 and compute a 95% confidence interval for each sample, approximately 95% of those intervals would contain the true population mean.

---

Would you like further details on the steps or any clarifications?

### Related Questions:
1. What is the relationship between confidence intervals and sample size?
2. How does increasing the confidence level affect the confidence interval width?
3. Why do we use Z-scores instead of T-scores in this case?
4. What are the implications of having a larger standard deviation in confidence interval estimation?
5. How would the confidence interval change if we had a larger sample size?

### Tip:
Always check whether the population standard deviation is known or unknown before deciding to use Z-scores or T-scores when constructing confidence intervals.

Learn how to calculate and interpret a 95% confidence interval for a population mean using a sample mean of 20, sample size of 25, and known standard deviation of 5. Follow the step-by-step process to apply Z-scores, standard error, and margin of error. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation