91Ó°ÊÓ

Let \(X\) be the outcome of a chance experiment with \(E(X)=\mu\) and \(V(X)=\) \(\sigma^{2}\). When \(\mu\) and \(\sigma^{2}\) are unknown, the statistician often estimates them by repeating the experiment \(n\) times with outcomes \(x_{1}, x_{2}, \ldots, x_{n},\) estimating \(\mu\) by the sample mean $$\bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}$$ and \(\sigma^{2}\) by the sample variance $$s^{2}=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}$$ Then \(s\) is the sample standard deviation. These formulas should remind the reader of the definitions of the theoretical mean and variance. (Many statisticians define the sample variance with the coefficient \(1 / n\) replaced by \(1 /(n-1)\). If this alternative definition is used, the expected value of \(s^{2}\) is equal to \(\sigma^{2}\). See Exercise \(18,\) part (d).) Write a computer program that will roll a die \(n\) times and compute the sample mean and sample variance. Repeat this experiment several times for \(n=10\) and \(n=1000\). How well do the sample mean and sample variance estimate the true mean \(7 / 2\) and variance \(35 / 12 ?\)

Step by step solution

01

Set Up the Environment

To solve this problem, we need to simulate rolling a die multiple times, for which we can use a programming language like Python. Be sure to have access to a Python environment, such as Jupyter Notebook or an IDE like PyCharm or VSCode.

02

Import Necessary Libraries

In Python, import libraries that will help simulate die rolls and perform calculations. Use 'import random' for random number generation and 'import numpy as np' for numeric calculations.

03

Define the Simulation Function

Create a function 'roll_die_and_compute_stats(n)' that simulates rolling a six-sided die 'n' times. Within this function, store the results of each roll in a list.

04

Calculate Sample Mean

Inside the function, calculate the sample mean. Use the formula: \( \bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i} \). In Python, this can be computed using 'mean = np.mean(rolls)'.

05

Calculate Sample Variance

Calculate the sample variance inside your function using the formula: \( s^{2}=\frac{1}{n} \sum_{i=1}^{n} (x_{i}-\bar{x})^{2} \). This can be computed in Python using 'variance = np.var(rolls)'.

06

Run the Simulation

Call your function multiple times with different values of 'n' to observe how estimations vary. Use 'n = 10' and 'n = 1000' for comparison.

07

Compare to True Mean and Variance

The theoretical mean is \( 3.5 \) and the variance is \( \frac{35}{12} \approx 2.9167 \). Compare your computed sample means and variances to these theoretical values to evaluate the accuracy of the estimates.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean

In statistics, the sample mean is a way to estimate the central tendency of a data set. Imagine you have several outcomes from an experiment, such as rolling a die multiple times. The sample mean helps you find the average outcome of those experiments. It's straightforward to calculate: sum up all your results and divide by the number of results.
To put it in a simple formula:

• If you named your results from the die rolls as \(x_1, x_2, \ldots, x_n\), the sample mean \(\bar{x}\) is calculated using \(\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\).

This average provides a single number that summarizes all your results. For a six-sided die, repeated rolls should hover around the number 3.5, which is the theoretical mean of an ideal, unbiased die.
By calculating the sample mean in simulations, you get an idea of how close your results are to the true average.

Standard Deviation

Standard deviation is a crucial concept in statistics, helping quantify the amount of variation or dispersion in a set of values. When you roll a die multiple times, you might not always roll each face an equal number of times. The standard deviation measures how spread out your rolls are from their average result.
Here's a simple breakdown:

• First, find out how much each roll deviates from the sample mean.
• Using these deviations, calculate the variance by averaging the squared differences.

In formula terms, first find the variance \(s^2\) using \(s^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\). Then, the standard deviation \(s\) is the square root of the variance, \(s = \sqrt{s^2}\).
The smaller the standard deviation, the closer your experimental rolls are to the average result, indicating consistent performance. A large standard deviation means more variability in your data.

Die Simulation

Simulating a die roll involves mimicking the behavior of a six-sided die digitally. It's a fantastic way to understand probability and statistics concepts hands-on. In our earlier example, the die has six possibilities for each roll: 1, 2, 3, 4, 5, or 6.
Simulating this could involve using a computer program or script to generate a random number between 1 and 6. By doing this multiple times, you record the outcomes as though you were physically rolling a die each time.
The benefits of a simulation include:

• Seeing statistical concepts in action, like sample mean and standard deviation.
• Comparisons between small and large sample sizes which reveal the impact on estimation accuracy.

When you simulate rolling a die, especially at different sample sizes like 10 or 1000 times, you can compare your results to theoretical averages and observe any discrepancies. This practice is essential in statistics because it illustrates how increasing the sample size can lead to more reliable estimates of the theoretical mean and variance. Through simulations, you become more comfortable with concepts like random variation and statistical inference.