Question

\(X\) sei eine Zufallsgröße vom Typ \(N(\mu, 1) \cdot\left[\gamma_{u}, \gamma_{o}\right]\) sei ein konkretes \(100(1-\alpha) \%\) Konfidenzintervall für den Erwartungswert \(\mu ; L=\gamma_{o}-\gamma_{u}\). (a) Beweisen Sie, dass die Sicherheitswahrscheinlichkeit \(1-\alpha\) wächst, wenn man den Stichprobenumfang verdoppelt, die Intervall-Länge \(L\) aber beibehält. Auf welchen Wert steigt speziell die Sicherheitswahrscheinlichkeit \(1-\alpha=0,90\) bei Verdoppelung des Stichprobenumfangs? (b) Um wieviel muss man den Stichprobenumfang erhöhen, wenn bei konstanter Sicherheitswahrscheinlichkeit die Länge \(L\) halbiert werden soll?

Short answer

(a) The confidence level increases to 95% with doubled sample size. (b) The sample size must increase fourfold to halve \(L\).

Step-by-step solution

Understand the Problem

The problem involves a random variable \(X\) with a normal distribution \(N(\mu, 1)\) and a given confidence interval \([\gamma_u, \gamma_o]\). You need to prove how the confidence level changes when the sample size is doubled while keeping the interval length \(L\) constant. Additionally, determine how much the sample size must increase to halve \(L\) with a constant confidence level.

Confidence Interval Formula

For a normal distribution with known variance, the \((1-\alpha)\times 100\%\) confidence interval for the mean \(\mu\) is given by \(\bar{x} \pm z_{1-\alpha/2} \times \frac{\sigma}{\sqrt{n}}\). Here, \(z_{1-\alpha/2}\) is the z-score corresponding to the confidence level, \(\sigma=1\) is the standard deviation, and \(n\) is the sample size.

Double the Sample Size

Increasing the sample size from \(n\) to \(2n\) reduces the standard error to \(\frac{1}{\sqrt{2}} \times \frac{\sigma}{\sqrt{n}}\). For a fixed interval length \(L\), this wider interval implies an increase in the z-score \(z'_{1-\alpha'/2}\), hence increasing the confidence level \(1-\alpha\).

Calculate Updated Confidence Level

To find the new confidence level when \(1-\alpha=0.90\), identify the z-score that corresponds for this confidence level initially, say \(z_{0.95}\) for a bilateral test. By doubling the sample size, adjust \(z'\) for the interval to remain \(L\): \(z'_{0.975} \approx 1.96\). The new confidence level will be higher than 90% based on the updated z-score possibly to a typical level like 95%.

Change in Sample Size to Halve Length

To achieve an interval of length \(\frac{L}{2}\) while maintaining the same confidence level, notice that the interval length is proportional to \(\frac{1}{\sqrt{n}}\). Halving \(L\) requires \(\frac{L}{2} = z_{0.95} \times \frac{1}{\sqrt{n'}}\). To solve for \(n'\), set it equal to \(4n\) since \(\frac{1}{2}\) is the square root of \(\frac{1}{n}\) requiring four times the original sample size.

Key concepts

Normal Distribution

The concept of normal distribution is central to statistics and vital in calculating confidence intervals. It is often represented as a bell-shaped curve, symmetric around the mean.
This distribution tells us how data values are expected to spread around the mean in the universe of possibilities.

• The mean (\( \mu \)) is the center of the distribution.
• The standard deviation (\( \sigma \)) indicates how much variation there is from the mean. In the problem, this value is 1, meaning the data points are closely centered around the mean.

The normal distribution is useful because it allows for straightforward calculations of probabilities and ranges within data sets, integral for deriving confidence intervals and interpreting statistical data. This concept is especially meaningful in large sample sizes as more data points align to the distribution pattern.

Sample Size

Sample size (\( n \)) is a crucial factor in determining the precision of statistical estimates like confidence intervals. Larger sample sizes narrow the confidence interval, providing higher accuracy about the population parameter.
This is because sample size inversely affects the standard error, which is the measure of variance in the data:
\[ \text{Standard Error} = \frac{\sigma}{\sqrt{n}} \]As you increase \( n \), the standard error decreases. Doubling the sample size results in a reduction of the standard error by a factor of \( \frac{1}{\sqrt{2}} \), which tightens the interval, increasing confidence levels.
Furthermore, changes to the sample size are necessary to adjust the confidence interval length and desired confidence level. In practice, determining the appropriate sample size requires balancing precision requirements with resource availability.

Statistical Significance

Statistical significance indicates whether a result is likely caused by something other than random chance. It is closely tied to the chosen confidence level \( 1-\alpha \).
A higher confidence level implies a higher degree of reliability that the parameter is accurately estimated.

• The confidence level reflects the probability that the interval so calculated will contain the true parameter value.
• By default, conventional significance levels are set at 5% (or less), resulting in a confidence level of 95% or more, meaning that in 95% of cases, the true parameter will fall within the interval.

For instance, increasing the sample size increases the confidence level from 90% to 95% because the wider confidence interval captures the actual population parameter more effectively, reducing the potential error.

Z-Score

A z-score is a statistical measurement that depicts the number of standard deviations a data point is from the mean. In the context of confidence intervals, the z-score determines the margin of error.

• A higher z-score indicates a broader margin, suited for higher confidence levels.
• Lower z-scores result in narrower margins, aligning with lower confidence levels.

The z-score is crucial for calculating confidence intervals for normally distributed data.
For a 95% confidence interval, a common z-score is 1.96, meaning the interval includes data that falls within 1.96 standard deviations from the mean on either side.
Adjusting the z-score according to exact confidence level requirements is necessary when modifying sample size either for tightening or relaxing the interval.