Problem 10 - Basiswissen Statistik: Kompaktkurs fur Anwender aus Wirtschaft, Informatik und Technik

Question

Geben sei eine normalverteilte Zufallsstichprobe vom Umfang \(n=20\), aus deren Realisation sich die Werte \(\sum_{i=1}^{n} x_{i}=100\) und \(S^{2}=10\) ergeben. Geben Sie ein Konfidenzintervall für den Erwartungswert zum Konfidenzniveau \(0.9\) an.

Step-by-Step Solution

Answer

The 90% confidence interval for the mean is \([3.777, 6.223]\).

Step 1: Understand the Problem

We need to calculate a confidence interval for the population mean \(\mu\) using a normally distributed random sample. Given are \(n=20\), \(\sum_{i=1}^{n} x_{i}=100\), and the sample variance \(S^{2}=10\). The confidence level is 90%.

Step 2: Calculate the Sample Mean

The sample mean \(\bar{x}\) is calculated by dividing the sum of the sample values by the number of observations. \[\bar{x} = \frac{\sum_{i=1}^{n} x_{i}}{n} = \frac{100}{20} = 5\]

Step 3: Determine the T-Critical Value

Since \(n\) is 20 and not large, we use the t-distribution. The degrees of freedom are \(n-1=19\). For a 90% confidence interval, we need \(\alpha=0.1\), so we find \(t_{\alpha/2,19}\) from a t-table or calculator. This value is approximately 1.729.

Step 4: Calculate the Standard Error

The standard error of the mean is calculated using the sample variance \(S^{2}\): \[\text{SE} = \sqrt{\frac{S^2}{n}} = \sqrt{\frac{10}{20}} = \sqrt{0.5} \approx 0.707\]

Step 5: Construct the Confidence Interval

The confidence interval is given by: \[\bar{x} \pm t_{\alpha/2,19} \times \text{SE} = 5 \pm 1.729 \times 0.707\] This calculates to \[5 \pm 1.223\] Therefore, the confidence interval is \([3.777, 6.223]\).

Key Concepts

Normal Distribution

A normal distribution is an important concept in statistics, often referred to as a bell curve because of its distinctive shape. It's symmetric and describes how values of a random variable are likely to be distributed around the mean. In normal distribution, the mean, median, and mode are all equal.

Some key properties include:

• 68% of the data falls within one standard deviation from the mean.
• 95% is within two standard deviations, and
• 99.7% is within three standard deviations.

Normal distributions are crucial because many natural phenomena fit this pattern. It allows statisticians to make inferences about a population based on sample data. In hypothesis testing and creating confidence intervals, the normal distribution often serves as a foundation for analysis.

T-Distribution

The t-distribution, also known as Student's t-distribution, is used when the sample size is small and the population standard deviation is unknown. It resembles the normal distribution but has fatter tails, which account for the additional variability expected with smaller samples.

Here’s how it works:

• The t-distribution becomes more like the normal distribution as sample size increases.
• Degrees of freedom (df), usually defined as the number of observations minus one, influence its shape.
• For our problem with a sample size of 20, we use df = 19.

In constructing confidence intervals with small samples, the t-distribution provides a critical value, denoted as \( t_{\alpha/2, df} \), that adjusts for uncertainty. This is essential when determining how broad the interval must be to achieve a certain level of confidence.

Standard Error

The standard error (SE) is a measure of how much the sample mean is likely to vary from the true population mean. It's calculated using the formula \( \text{SE} = \sqrt{\frac{S^2}{n}} \), where \( S^2 \) is the sample variance and \( n \) is the sample size.

The standard error gives insight into the precision of the sample mean as an estimate of the population mean:

• Smaller SE indicates more reliable estimates.
• Larger samples typically have smaller SE, increasing precision.

In our example, the standard error was computed to be approximately 0.707, highlighting the expected variability in sample means around the population mean. Understanding SE is crucial for constructing narrow and accurate confidence intervals.

Sample Mean

The sample mean, usually denoted as \( \bar{x} \), is the average of all numbers in a sample. It's calculated by dividing the sum of sample values by the number of observations. The sample mean serves as an unbiased estimator of the population mean under the assumption of random sampling.

In practice:

• Use \( \bar{x} = \frac{\sum x_i}{n} \) to find the sample mean.
• Acts as the center point in confidence interval calculations.
• Helps identify the distribution's central tendency.

In the given scenario, the sample mean was calculated as 5. This value is used as the midpoint of our confidence interval, a crucial step in estimating where the true population mean lies with a certain level of confidence.