If you wish to estimate a population mean with a sampling distribution error SE=0.34 using a​ 95% confidence interval and you know from prior sampling that sigma squared is approximately equal to 8.2​, how many observations would have to be included in your​ sample?

To estimate the number of observations (sample size \(n\)) required for a 95% confidence interval with a given standard error, we can use the formula for the standard error (SE) of the sample mean:

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

Where:
- \( SE \) is the standard error,
- \( \sigma \) is the population standard deviation,
- \( n \) is the sample size.

Given:
- \( SE = 0.34 \),
- \( \sigma^2 = 8.2 \) (so \( \sigma = \sqrt{8.2} \)),
- Confidence level = 95%.

Step 1: Calculate \( \sigma \).
\[
\sigma = \sqrt{8.2} \approx 2.863
\]

Step 2: Rearrange the standard error formula to solve for \( n \).
\[
n = \left( \frac{\sigma}{SE} \right)^2
\]

Substitute the known values:
\[
n = \left( \frac{2.863}{0.34} \right)^2
\]

Now let's calculate \( n \).The required sample size \( n \) is approximately 71 observations to estimate the population mean with a standard error of 0.34 and a 95% confidence interval.

Would you like further details on how this was calculated or have any additional questions?

Here are 5 related questions:
1. What is the relationship between confidence intervals and sample size?
2. How would the required sample size change if we used a 99% confidence interval?
3. What is the impact of increasing the standard error on the sample size?
4. How is the confidence interval affected if the population variance increases?
5. How does the sample size formula differ for proportions versus means?

**Tip:** Increasing your confidence level will generally require a larger sample size to maintain the same precision.

If you wish to estimate a population mean with a sampling distribution error SE=0.34 using a 95% confidence interval and you know from prior sampling that sigma squared is approximately equal to 8.2, how many observations would have to be included in your sample?

Learn how to calculate the sample size needed to estimate a population mean with a standard error of 0.34 and a 95% confidence interval. This problem involves key statistical concepts such as the standard error formula and the impact of population variance on sample size.

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