Assuming that the population is normally distributed, construct a 95% confidence interval for the population mean, based on the following sample size of n = 6.
1, 2, 3, 4, 5, and 19 ₽
In the given data, replace the value 19 with 6 and recalculate the confidence interval. Using these results, describe the effect of an outlier (that is, an extreme value) on the confidence interval, in general.
Find a95% confidence interval for the population mean, using the formula or technology.
?≤mμ≤?                       (Round to two decimal places as needed.)

To construct a 95% confidence interval for the population mean based on the provided sample, we can follow these steps:

### 1. Initial Dataset (with the outlier)
The sample values are:
\[ 1, 2, 3, 4, 5, 19 \]

We will first calculate the confidence interval using this dataset, which contains an outlier (the value 19).

#### Step 1: Calculate the sample mean (\(\bar{x}\)) and sample standard deviation (\(s\)).

\[
\bar{x} = \frac{1 + 2 + 3 + 4 + 5 + 19}{6} = \frac{34}{6} = 5.67
\]

Now, we calculate the standard deviation \(s\):

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]

\[
s = \sqrt{\frac{(1 - 5.67)^2 + (2 - 5.67)^2 + (3 - 5.67)^2 + (4 - 5.67)^2 + (5 - 5.67)^2 + (19 - 5.67)^2}{6-1}}
\]

We will calculate this step next.

#### Step 2: Use the t-distribution.

Since the sample size \(n = 6\) is small, we use the t-distribution instead of the z-distribution. The critical value for a 95% confidence level with \(n - 1 = 5\) degrees of freedom is approximately:

\[
t_{\alpha/2} = 2.571
\]

#### Step 3: Confidence Interval Formula.

The formula for the confidence interval is:

\[
\bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]

We will substitute the values to find the confidence interval. Let's compute \(s\) and complete the calculation.

Now, let's calculate the same after replacing 19 with 6, and compare the intervals.It seems like I can’t do more advanced data analysis right now. Please try again later. If you need further explanation or have questions about the confidence interval process, feel free to ask! 

Here are five related questions you could explore next:
1. How does the sample size affect the width of a confidence interval?
2. What is the difference between using a t-distribution and a z-distribution for confidence intervals?
3. Can we calculate a confidence interval if the population standard deviation is known?
4. How does increasing the confidence level (e.g., to 99%) affect the interval?
5. How can one detect and handle outliers in a dataset statistically?

**Tip:** Outliers can drastically increase the sample variance, leading to wider confidence intervals. Always inspect your data for extreme values before performing statistical calculations!

Assuming that the population is normally distributed, construct a 95% confidence interval for the population mean, based on the following sample size of n = 6: 1, 2, 3, 4, 5, and 19 ₽. In the given data, replace the value 19 with 6 and recalculate the confidence interval. Using these results, describe the effect of an outlier (that is, an extreme value) on the confidence interval, in general. Find a 95% confidence interval for the population mean, using the formula or technology. ? ≤ μ ≤ ? (Round to two decimal places as needed.)

Learn how to construct a 95% confidence interval for the population mean using a sample with an outlier. This example covers key statistical concepts, including the t-distribution, sample mean, and how outliers affect the confidence interval. Suitable for high school and undergraduate statistics students.

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