Problem 2
Question
a) Bestimmen Sie mit Hilfe der Ihnen zur Verfügung stehenden Tabellen folgende Quantile der \(N(0,1)-, \chi^{2}\) - bzw. der \(t\)-Verteilung: $$ \chi_{7 ; 0.975}^{2} \quad \chi_{120 ; 0.95}^{2} \quad t_{29 ; 0.01} \quad t_{180 ; 0.975} $$ Hinweis: Nutzen Sie gegebenenfalls die Symmetrie der Normalbzw. \(t\)-Verteilung. Beachten Sie außerdem die für große \(n\) gültigen Näherungsformeln \(\chi_{n ; p}^{2} \approx n+u_{p} \sqrt{2 n}\) und \(t_{n ; p} \approx u_{p}\). b) Die Zufallsvariablen \(X_{1}, \ldots, X_{100}\) seien unabhängig und identisch ( \(N(0,1)\)-verteilt. Bestimmen Sie Zahlen \(s_{1}, s_{2}\) und \(s_{3}\) so, dass gilt: (i) \(P\left(X_{1}^{2}+\cdots+X_{20}^{2} \geq s_{1}\right)=0.05\) (ii) \(P\left(X_{29}^{2} /\left(X_{1}^{2}+\cdots+X_{28}^{2}\right) \leq s_{2}\right)=0.95\) (iii) \(P\left(\left|X_{1}+\cdots+X_{100}\right| / 100 \leq s_{3}\right)=0.95\).
Step-by-Step Solution
Verified Answer
a) \(\chi_{7; 0.975}^{2} = 2.167\, \(\chi_{120; 0.95}^{2} = 146.566\, \(t_{29; 0.01} = -2.462\, \(t_{180; 0.975} = 1.972\. b) \(s_{1} = 31.41\, \(s_{2} = 4.20\, \(s_{3} = 2.575\.
1Step 1: Identify the given quantiles
We need to find the following quantiles using the provided tables:i) \(\chi^{2}_{\cdot}\) quantiles:- \(\chi_{7; 0.975}^{2}\)- \(\chi_{120; 0.95}^{2}\)ii) \(t_{\cdot}\) quantiles:- \(t_{29; 0.01}\)- \(t_{180; 0.975}\)
2Step 2: Determine \(\chi_{7; 0.975}^{2}\)
Refer to the \(\chi^{2}\) table for degrees of freedom (df) = 7 and a probability of 0.975. The corresponding \(\chi_{7; 0.975}^{2}\) value is approximately 2.167.
3Step 3: Determine \(\chi_{120; 0.95}^{2}\)
Refer to the \(\chi^{2}\) table for degrees of freedom (df) = 120 and a probability of 0.95. The corresponding \(\chi_{120; 0.95}^{2}\) value is approximately 146.566.
4Step 4: Determine \(t_{29; 0.01}\)
Refer to the Student's t-distribution table for degrees of freedom (df) = 29 and a probability of 0.01. The corresponding \(t_{29; 0.01}\) value is approximately -2.462.
5Step 5: Determine \(t_{180; 0.975}\)
Refer to the Student's t-distribution table for degrees of freedom (df) = 180 and a probability of 0.975. The corresponding \(t_{180; 0.975}\) value is approximately 1.972.
6Step 6.1: Calculate \(s_{1}\)
Given \(X_{1}^2 + \cdots + X_{20}^2 \sim \chi^{2} ___ {20}\) and \(P(\chi^{2} ___ {20} \geq \ s_{1}) = 0.05\), using the Chi-square table, \(s_{1}\) is approximately 31.41.
7Step 6.2: Calculate \(s_{2}\)
Given \(\frac{X_{29}^{2}}{X_{1}^{2} + \cdots + X_{28}^{2}} \sim F(1, 28)\) and \(P(F(1, 28) \leq \ s_{2}) = 0.95\), using the F-distribution table, \(s_{2}\) value is approximately 4.20.
8Step 6.3: Calculate \(s_{3}\)
Since \(X_{1} + \cdots + X_{100}\) is normally distributed with mean 0 and variance 100, the distribution \(\frac{X_{1} + \cdots + X_{100}}{10} \sim N(0,1)\). Therefore, \(P(\abs{\frac{S_{100}}{10}} \leq \ s_{3}) = 0.95\), using the standard normal table for middle 95% range, \(s_{3}\) is approximately 2.575.
Key Concepts
Quantile DeterminationChi-Square DistributionStudent's t-DistributionProbability CalculationF-Distribution
Quantile Determination
Quantile determination involves finding a value, called a quantile, from a specified distribution such that a certain fraction of the distribution's probability lies below this value. Quantiles are essential in statistical analysis, simplifying complex data sets into understandable thresholds.
To understand quantile determination, we need to refer to various statistical tables, such as those for the Chi-Square, t-Distribution, and F-Distribution. Here's a brief breakdown:
- The Chi-Square table helps us find values for \( \chi^{2} \) distributions given degrees of freedom and a probability level.
- The t-Distribution table is used to determine the t-scores critical to the t-tests, involving sample means with unknown variances.
- The F-Distribution table provides critical values for the F-Statistic, helping in variance comparison between groups.
To understand quantile determination, we need to refer to various statistical tables, such as those for the Chi-Square, t-Distribution, and F-Distribution. Here's a brief breakdown:
- The Chi-Square table helps us find values for \( \chi^{2} \) distributions given degrees of freedom and a probability level.
- The t-Distribution table is used to determine the t-scores critical to the t-tests, involving sample means with unknown variances.
- The F-Distribution table provides critical values for the F-Statistic, helping in variance comparison between groups.
Chi-Square Distribution
The Chi-Square distribution, denoted as \( \chi^{2} \), is crucial for hypothesis testing and variance analysis. It's particularly useful for goodness-of-fit tests and tests of independence. It is defined by its degrees of freedom (df), which usually correspond to the number of variables minus one.
To find a chi-square quantile such as \( \chi^{2}\textsubscript{7, 0.975}\textsuperscript{2} \), refer to the chi-square table at df = 7 and a probability value of 0.975. The corresponding value is approximately 2.167.
To find a chi-square quantile such as \( \chi^{2}\textsubscript{7, 0.975}\textsuperscript{2} \), refer to the chi-square table at df = 7 and a probability value of 0.975. The corresponding value is approximately 2.167.
Student's t-Distribution
Student's t-Distribution, or simply the t-distribution, is utilized in situations where the sample size is small and/or population variance is unknown. It helps in estimating population parameters when sample sizes are n < 30.
Accessing the t-Distribution table, you can find quantiles based on degrees of freedom and specific probability levels. For instance, to find \( t_{29 ; 0.01} \), look at the t-table for df = 29 at a significance level of 0.01. The value you'll find is around -2.462.
Accessing the t-Distribution table, you can find quantiles based on degrees of freedom and specific probability levels. For instance, to find \( t_{29 ; 0.01} \), look at the t-table for df = 29 at a significance level of 0.01. The value you'll find is around -2.462.
Probability Calculation
Probability Calculation involves determining the likelihood of a given outcome within a statistical distribution. This is central to forming predictions and making decisions based on data.
For example, when given that the variables \( X_{1}, \ldots, X_{20} \) are independently and identically \( N(0,1) \)-distributed, we would calculate probabilities such as \( P(X_1^2 + \ldots + X_{20}^2 \ge s_{1}) = 0.05 \). By using the chi-square distribution table, you find \( s_1 \) to be approximately 31.41.
For example, when given that the variables \( X_{1}, \ldots, X_{20} \) are independently and identically \( N(0,1) \)-distributed, we would calculate probabilities such as \( P(X_1^2 + \ldots + X_{20}^2 \ge s_{1}) = 0.05 \). By using the chi-square distribution table, you find \( s_1 \) to be approximately 31.41.
F-Distribution
The F-Distribution is utilized primarily in analysis of variance (ANOVA) and in regression analysis. It compares two variances to ascertain whether they significantly differ from each other.
The F-Distribution table aids in determining critical values. For instance, \( \frac{X_{29}^{2}}{X_{1}^{2} + \ldots + X_{28}^{2}} \sim F(1, 28) \) and with 95% probability, the critical value from the F-distribution table is roughly 4.20.
The F-Distribution table aids in determining critical values. For instance, \( \frac{X_{29}^{2}}{X_{1}^{2} + \ldots + X_{28}^{2}} \sim F(1, 28) \) and with 95% probability, the critical value from the F-distribution table is roughly 4.20.