Certainly! Below is an example of how to interpret a confidence interval, based on hypothetical data values.

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**Interpretation of Confidence Interval**

**Given Values:**
– Sample Mean (\(\bar{x}\)): 75
– Standard Error (SE): 5
– Confidence Level: 95%

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### **Step 1: Understanding the Confidence Interval Formula**

A confidence interval (CI) is a range of values that is used to estimate an unknown population parameter. The formula for a confidence interval is:

\[
\text{CI} = \bar{x} \pm Z \times \text{SE}
\]

Where:
– \(\bar{x}\) is the sample mean,
– \(Z\) is the Z-score corresponding to the chosen confidence level (for 95% confidence, \(Z = 1.96\)),
– \(\text{SE}\) is the standard error of the sample mean.

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### **Step 2: Calculation of the Confidence Interval**

For a 95% confidence level:
– The Z-score is **1.96**.
– Sample Mean (\(\bar{x}\)) = **75**
– Standard Error (SE) = **5**

Now, calculate the margin of error:

\[
\text{Margin of Error} = 1.96 \times 5 = 9.8
\]

Thus, the 95% confidence interval is:

\[
\text{CI} = 75 \pm 9.8
\]

This results in:

\[
\text{CI} = (65.2, 84.8)
\]

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### **Step 3: Interpretation**

The **95% confidence interval** for the population mean is **(65.2, 84.8)**. This means that we are **95% confident** that the true population mean lies between 65.2 and 84.8 based on the sample data.

– **Confidence Level:** The 95% confidence level indicates that if we were to take 100 different samples and compute a confidence interval for each sample, approximately 95 of those intervals would contain the true population mean.
– **Interpretation of Range:** The true population mean is likely to be within the range of 65.2 to 84.8, but there is a 5% chance that the true mean lies outside of this range.

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### **Conclusion:**

In conclusion, the confidence interval of **(65.2, 84.8)** provides an estimated range for the population mean based on the sample data. The interpretation suggests a high level of certainty about where the true mean is located, and the interval provides a useful tool for making informed decisions in the context of statistical inference.

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This explanation breaks down the process of calculating and interpreting a confidence interval, ensuring clarity and accuracy in a technical context.