To answer the question and estimate a 95% confidence interval for the true mean of the population, we'll follow these steps:

Step 1: Understand the Data

You have a sample of 100 students selected from a population of 700 students, and based on the sample, we need to estimate a confidence interval for the population mean.

Given data from your sample:

• Sum of Samples: 7408
• Number of Samples: 100

Step 2: Calculate the Sample Mean and Sample Standard Deviation

Sample Mean (x̄): $\text{Sample Mean} = \frac{\text{Sum of Sample Values}}{\text{Number of Samples}} = \frac{7408}{100} = 74.08$

Now, we would need the sample standard deviation (s) to proceed with the next step. Since the question suggests using a statistics calculator, let's assume that this value has been calculated based on the sample.

For this example, let's assume the sample standard deviation (s) is 5.5 (you can replace this with the actual value from your statistics calculator if available).

Step 3: Use the Normal Distribution/Empirical Rule to Estimate the 95% Confidence Interval

The 95% confidence interval for the population mean can be estimated using the formula: $\text{Confidence Interval} = \text{Sample Mean} \pm Z \times \frac{s}{\sqrt{n}}$ Where:

• Z = 1.96 (This is the Z-value corresponding to a 95% confidence interval in a standard normal distribution)
• s = Sample standard deviation (5.5 in this example)
• n = Sample size (100)

Step 3.1: Calculate the Standard Error (SE)

$\text{Standard Error (SE)} = \frac{s}{\sqrt{n}} = \frac{5.5}{\sqrt{100}} = \frac{5.5}{10} = 0.55$

Step 3.2: Compute the Margin of Error (MOE)

$\text{Margin of Error (MOE)} = Z \times \text{SE} = 1.96 \times 0.55 = 1.078$

Step 3.3: Calculate the Confidence Interval

$\text{Confidence Interval} = 74.08 \pm 1.078$ This gives: $\text{Lower Bound} = 74.08 - 1.078 = 73.0$ $\text{Upper Bound} = 74.08 + 1.078 = 75.2$

Step 4: Conclusion

Thus, the 95% confidence interval for the population mean is approximately (73.0, 75.2).

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Would you like to explore more about this process, such as the impact of changing sample size or standard deviation? Let me know if you need clarification on any steps!

Further Questions to Consider:

1. How does increasing the sample size affect the confidence interval?
2. What happens if the sample standard deviation is much larger than expected?
3. How would you calculate the confidence interval if you had a population standard deviation instead of a sample standard deviation?
4. What is the difference between a confidence interval and a prediction interval?
5. How can you use a t-distribution instead of a normal distribution in this case?

Tip:

For greater accuracy in estimating the population mean, try to use a larger sample size. A larger sample tends to give a more precise estimate and narrows the confidence interval.