Question

. To illustrate Exercise 5.4.22, let \(X_{1}, X_{2}, \ldots, X_{9}\) and \(Y_{1}, Y_{2}, \ldots, Y_{12}\) represent two independent random samples from the respective normal distributions \(N\left(\mu_{1}, \sigma_{1}^{2}\right)\) and \(N\left(\mu_{2}, \sigma_{2}^{2}\right) .\) It is given that \(\sigma_{1}^{2}=3 \sigma_{2}^{2}\), but \(\sigma_{2}^{2}\) is unknown. Define a random variable which has a \(t\) -distribution that can be used to find a 95 percent confidence interval for \(\mu_{1}-\mu_{2}\).

Short answer

The formulated random variable which takes a t-distribution is \(Z = (\bar{X} - \bar{Y}) / \mathrm{SE} (\bar{X} - \bar{Y})\) with 8 degrees of freedom. The 95% confidence interval for \(\mu_{1}-\mu_{2}\) is given by \((\bar{X} - \bar{Y}) \pm t_{\alpha/2,df} . \mathrm{SE} (\bar{X} - \bar{Y})\)

Step-by-step solution

Formulate the Random Variable

The random variable \(Z\) is defined as \((\bar{X} - \bar{Y}) / \mathrm{SE} (\bar{X} - \bar{Y})\). Where, \(\bar{X}\) and \(\bar{Y}\) are the sample means, and \(\mathrm{SE} (\bar{X} - \bar{Y})\) is the standard error of the difference in means. Considering the fact that \(\sigma_{1}^{2}=3\sigma_{2}^{2}\), both variances are equal to say \(\sigma^{2}\). Thus, \(\mathrm{SE} (\bar{X} - \bar{Y})\) square will be equal to \(\sigma^{2} (1/n_1 + 1/n_2)\), ranging over the sample size for each sample.

Identify the Degree of Freedom and the t-Distribution

The degree of freedom (df) for a two-sample t-test is given by the minimum of \(n_1 - 1\) and \(n_2 - 1\) where \(n_1\) and \(n_2\) are the sample sizes. Here, the sample sizes are 9 and 12, so the degrees of freedom would be the minimum between 8 and 11, which is 8. So, the random variable \(Z\) takes a t-distribution with 8 degrees of freedom.

Formulate the Confidence Interval

Finally, a 95% confidence interval for \(\mu_{1}-\mu_{2}\) is defined by \((\bar{X} - \bar{Y}) \pm t_{\alpha/2,df} . \mathrm{SE} (\bar{X} - \bar{Y})\), where \(t_{\alpha/2,df}\) is the t-value for a 95% confidence interval with \(df\) degrees of freedom, to be obtained from the t-distribution table.

Key concepts

Random Samples

Random samples are a fundamental concept in statistics. They represent subsets of data that are drawn from an overall population, where each member of the population has an equal chance of being included in the sample. This equality of selection ensures that the sample is a representative snapshot of the entire population.
In the context of our exercise, we have two independent random samples. The first sample is comprised of 9 observations, denoted as \(X_1, X_2, \ldots, X_9\), from a normal distribution \(N(\mu_1, \sigma_1^2)\). The second sample includes 12 observations, denoted as \(Y_1, Y_2, \ldots, Y_{12}\), from another normal distribution \(N(\mu_2, \sigma_2^2)\).
These random samples will help us estimate the parameters of the population, specifically the means \(\mu_1\) and \(\mu_2\), and evaluate the differences between them. Such samples are crucial for statistical inference, as they allow us to generalize our findings to a larger group beyond the samples themselves.

Confidence Interval

A confidence interval is a range of values that is likely to contain a population parameter, such as a mean, with a specified level of confidence. It gives us an estimate of the unknown parameter based on our sample data.
In our exercise, we're focusing on finding a 95% confidence interval for the difference between two population means, \(\mu_1 - \mu_2\). This means that we are 95% confident that the actual difference between the population means lies within our calculated interval.
To calculate this interval, we use the formula:

• \((\bar{X} - \bar{Y}) \pm t_{\alpha/2, df} \times \text{SE}(\bar{X} - \bar{Y}) \)

Here, \(\bar{X} - \bar{Y}\) is the difference between the sample means, \(t_{\alpha/2, df}\) is the t-value from the t-distribution, and \(\text{SE}(\bar{X} - \bar{Y})\) is the standard error of the difference in means. The t-value depends on our desired level of confidence (95% in this case) and the degrees of freedom. The confidence interval provides a useful way to understand the statistical uncertainty in our estimates.

Degree of Freedom

Degrees of freedom (df) refer to the number of independent values or quantities that can be assigned within a statistical calculation. In general, they reflect the number of values that are free to vary, giving us flexibility in statistical testing.
In the context of our exercise, when conducting a two-sample t-test, the degrees of freedom are determined by the sample sizes of our two independent samples. Specifically, df is calculated as the minimum of \((n_1 - 1)\) and \((n_2 - 1)\). Here, \(n_1\) is 9 and \(n_2\) is 12, leading to degrees of freedom being the smaller of 8 and 11, which is 8.
The degrees of freedom influence the shape of the t-distribution, which is slightly different from a normal distribution, especially with smaller sample sizes. They are crucial for determining the appropriate critical t-value when constructing confidence intervals. In our example, a t-distribution with 8 degrees of freedom is used to find the t-value necessary for constructing the 95% confidence interval. Understanding how df work helps improve the reliability of our statistical estimates.