Problem 12 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Let \(Z_{1}, \ldots, Z_{n}\) be a random sample from an \(N(0,1)\) distribution. Define \(X_{i}=\mu+\sigma Z_{i}\) for \(i=1, \ldots, n\) and \(\sigma>0\). Let \(\bar{Z}, \bar{X}\) denote the sample averages and \(S_{Z}\) and \(S_{X}\) the sample standard deviations, of the \(Z_{i}\) and \(X_{i}\), respectively. a. Show that \(X_{1}, \ldots, X_{n}\) is a random sample from an \(N\left(\mu, \sigma^{2}\right)\) distribution. b. Express \(\bar{X}\) and \(S_{X}\) in terms of \(\bar{Z}, S_{Z}, \mu\), and \(\sigma\). c. Verify that $$ \frac{\bar{X}-\mu}{S_{X} / \sqrt{n}}=\frac{\bar{Z}}{S_{Z} / \sqrt{n}} $$ and explain why this shows that the distribution of the studentized mean does not depend on \(\mu\) and \(\sigma\).

Step-by-Step Solution

Answer

The studentized mean \(\frac{\bar{X}-\mu}{S_{X}/\sqrt{n}}\) equals \(\frac{\bar{Z}}{S_{Z}/\sqrt{n}}\), showing it does not depend on \(\mu\) and \(\sigma\).

Step 1: Show the distribution of X_i

Since \(Z_i \sim N(0,1)\) and \(X_i = \mu + \sigma Z_i\), we can express \(X_i\) as a linear transformation of \(Z_i\). The linear transformation of a normal distribution \(N(\mu, \sigma^2)\) has the following rules: \[\begin{align*} E[X_i] &= E[\mu + \sigma Z_i] = \mu + \sigma E[Z_i] = \mu, \ \text{Var}(X_i) &= \text{Var}(\mu + \sigma Z_i) = \sigma^2 \text{Var}(Z_i) = \sigma^2. \end{align*}\]Therefore, \(X_i \sim N(\mu, \sigma^2)\).

Step 2: Express \(\bar{X}\) in terms of \(\bar{Z}\)

The sample average \(\bar{X}\) is defined as \[\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i = \frac{1}{n} \sum_{i=1}^n (\mu + \sigma Z_i).\]This can be simplified as \[\bar{X} = \mu + \sigma \left(\frac{1}{n} \sum_{i=1}^n Z_i\right) = \mu + \sigma \bar{Z}.\]

Step 3: Express \(S_{X}\) in terms of \(S_{Z}\)

The sample standard deviation \(S_X\) is \[S_X = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2}.\]Since \(X_i = \mu + \sigma Z_i\), it follows \[S_X = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (\sigma Z_i - \sigma \bar{Z})^2} = \sigma \sqrt{\frac{1}{n-1}\sum_{i=1}^n (Z_i - \bar{Z})^2} = \sigma S_Z.\]

Step 4: Verify the studentized mean identity

We need to show that \[\frac{\bar{X}-\mu}{S_{X}/\sqrt{n}} = \frac{\bar{Z}}{S_{Z}/\sqrt{n}}.\]Using the expressions derived above, substitute \( \bar{X} = \mu + \sigma \bar{Z} \) and \( S_X = \sigma S_Z \):\[\frac{\bar{X} - \mu}{S_X / \sqrt{n}} = \frac{(\mu + \sigma \bar{Z}) - \mu}{\sigma S_Z / \sqrt{n}} = \frac{\sigma \bar{Z}}{\sigma S_Z / \sqrt{n}} = \frac{\bar{Z}}{S_Z / \sqrt{n}}.\]This identity confirms that the studentized mean is independent of \(\mu\) and \(\sigma\).

Key Concepts

Normal Distribution

The normal distribution is a fundamental concept in probability theory. It describes how data points are distributed around a mean value. It is symmetric and has a bell-shaped curve, characterized by its mean \( \mu \,\) and variance \( \sigma^2 \,\). The standard form of a normal distribution is denoted as \( N(0,1) \,\), which means a mean of zero and a variance of one. This is often called the "standard normal distribution." The transformation from a standard normal distribution to any normal distribution involves scaling by the standard deviation \( \sigma \,\) and shifting by the mean \( \mu \,\). Understanding this distribution is pivotal as it serves as a foundation for many statistical methods and tests.

Some key properties of the normal distribution include:

• Approximately 68% of the data falls within one standard deviation of the mean.
• About 95% fall within two standard deviations.
• Nearly 99.7% lie within three standard deviations.

This makes it very useful in statistical forecasting and hypothesis testing.

Random Sample

A random sample is a set of observations drawn from a population where each individual has an equal chance of being selected. This means that every selection is independent of others, helping to ensure that the sample represents the entire population without bias.

In mathematical terms, a random sample is treated as a random variable, which can be used to make inferences about the population.

• In the given exercise, \( Z_1, \ldots, Z_n \,\) represents a random sample drawn from a standard normal distribution \( N(0,1) \,\).
• This randomness is critical as it allows the use of statistical methods to generalize findings from the sample to the broader population.

Random sampling is common in confidence interval estimation and hypothesis testing, ensuring that findings are valid and not due to sampling bias.

Student's t-distribution

The Student's t-distribution is utilized when dealing with small sample sizes or when the population standard deviation is unknown. It is similar in shape to the normal distribution but has heavier tails, meaning it is more prone to producing values that fall far from its mean.

This distribution becomes crucial when working with the sample mean and standard deviation in small samples. As sample size increases, the t-distribution approaches the normal distribution, making it highly reliable.

• In the context of the exercise, the transformation \( \frac{\bar{X} - \mu}{S_X / \sqrt{n}} = \frac{\bar{Z}}{S_Z / \sqrt{n}} \,\) could be seen as aligning with concepts from the t-distribution when the population parameters are unknown.
• This approach helps maintain accuracy in estimation and inference under uncertainty of population parameters.

Understanding this distribution is vital for tackling problems involving small samples, offering a method to control for sample variability.

Sample Mean

The sample mean is an average calculated from data points in a sample. It serves as an estimator of the population mean. In statistics, one typically denotes the sample mean as \( \bar{X} \,\) for the variable \( X \,\), which in this exercise is derived from the transformation of normal variables.

The sample mean is calculated using the formula:
\[ \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \]
where \( n \,\) is the number of observations in the sample.

• For the exercise, the relationship is given by \( \bar{X} = \mu + \sigma \bar{Z} \,\), linking the sample mean to the transformation parameters.

The benefit of using the sample mean is that it provides a central measure, helping in estimating and making predictions about the population from which the sample is drawn.

Sample Standard Deviation

The sample standard deviation is a measure that quantifies the amount of variation or dispersion of a set of data points. It is represented by \( S_X \,\) for a variable \( X \,\) and offers insight into the variability within a sample.

It is computed as:
\[ S_X = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2} \]
where \( \bar{X} \,\) is the sample mean.

• The exercise demonstrates that \( S_X = \sigma S_Z \,\), illustrating how the transformations affect data variability.

Using the sample standard deviation helps in comparing the spread between different datasets and is integral in hypothesis testing and confidence interval estimation. It helps understand how much individual data points deviate from the sample mean, which is crucial for interpreting the reliability and significance of data.