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Given \(x_{1}\) and \(x_{2}\) distributions that are normal or approximately normal with unknown \(\sigma_{1}\) and \(\sigma_{2}\), the value of \(t\) corresponding to \(\bar{x}_{1}-\bar{x}_{2}\) has a distribution that is approximated by a Student's \(t\) distribution. We use the convention that the degrees of freedom is approximately the smaller of \(n_{1}-1\) and \(n_{2}-1\). However, a more accurate estimate for the appropriate degrees of freedom is given by Satterthwaite's formula: $$\text { d.f. } \approx \frac{\left(\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}\right)^{2}}{\frac{1}{n_{1}-1}\left(\frac{s_{1}^{2}}{n_{1}}\right)^{2}+\frac{1}{n_{2}-1}\left(\frac{s_{2}^{2}}{n_{2}}\right)^{2}}$$ where \(s_{1}, s_{2}, n_{1}\), and \(n_{2}\) are the respective sample standard deviations and sample sizes of independent random samples from the \(x_{1}\) and \(x_{2}\) distributions. This is the approximation used by most statistical software. When both \(n_{1}\) and \(n_{2}\) are 5 or larger, it is quite accurate. The degrees of freedom computed from this formula are either truncated or not rounded. (a) In Problem 13, we tested whether the population average crime rate \(\mu_{2}\) in the Rocky Mountain region is higher than that in New England, \(\mu_{1}\). The data were \(n_{1}=10, \bar{x}_{1} \approx 3.51, s_{1} \approx 0.81, n_{2}=12, \bar{x}_{2} \approx 3.87\), and \(s_{2} \approx 0.94\). Use Satterthwaite's formula to compute the degrees of freedom for the Student's \(t\) distribution. (b) When you did Problem 13 , you followed the convention that degrees of freedom \(d . f .=\) smaller of \(n_{1}-1\) and \(n_{2}-1 .\) Compare this value of \(d . f\). with that found with Satterthwaite's formula.

Step by step solution

01

Understand the Formula

The degrees of freedom (d.f.) using Satterthwaite's formula is calculated as: \[\text{d.f.} \approx \frac{\left(\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}\right)^{2}}{\frac{1}{n_{1}-1}\left(\frac{s_{1}^{2}}{n_{1}}\right)^{2}+\frac{1}{n_{2}-1}\left(\frac{s_{2}^{2}}{n_{2}}\right)^{2}}\] Given values are \(s_{1} = 0.81\), \(n_{1} = 10\), \(s_{2} = 0.94\), and \(n_{2} = 12\).

02

Calculate Numerator

First compute the numerator of the formula: \[\left(\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}\right)^2 = \left(\frac{(0.81)^2}{10} + \frac{(0.94)^2}{12}\right)^2\]Calculate each part, starting with:1. \((0.81)^2 = 0.6561\).2. \((0.94)^2 = 0.8836\).3. Combine: \(\frac{0.6561}{10} + \frac{0.8836}{12} = 0.06561 + 0.07363 = 0.13924\).4. Square \(0.13924\): \((0.13924)^2 = 0.01939\).

03

Calculate Denominator

Now compute the denominator of Satterthwaite's formula:\[\frac{1}{n_{1}-1}\left(\frac{s_{1}^{2}}{n_{1}}\right)^{2} + \frac{1}{n_{2}-1}\left(\frac{s_{2}^{2}}{n_{2}}\right)^{2}\]1. \(\left(\frac{s_{1}^{2}}{n_{1}}\right)^2 = (0.06561)^2 = 0.004309\).2. \(\left(\frac{s_{2}^{2}}{n_{2}}\right)^2 = (0.07363)^2 = 0.005424\).3. Divide by respective degrees minus 1: - \(\frac{0.004309}{9} = 0.000479\) - \(\frac{0.005424}{11} = 0.000493\)Add these two parts: \(0.000479 + 0.000493 = 0.000972\).

04

Compute Satterthwaite's Degrees of Freedom

Divide the numerator by the denominator:\[\text{d.f.} \approx \frac{0.01939}{0.000972} \approx 19.95\]The degrees of freedom, according to Satterthwaite's formula, are approximately 19.95, which is typically truncated to 19.

05

Calculate Degrees of Freedom Conventionally

According to the conventional approach, you take the smaller of \(n_{1} - 1\) and \(n_{2} - 1\):\[\text{d.f.} = \min(10 - 1, 12 - 1) = \min(9, 11) = 9\]

06

Compare the Degrees of Freedom

The degrees of freedom calculated using Satterthwaite's formula is approximately 19.95 (truncated to 19), compared to the conventional approach, which gives a value of 9. Satterthwaite accounts for more variability by giving a higher d.f. value.

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Key Concepts

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

Student's t distribution

The Student's t distribution is a probability distribution used to assess the differences between sample means, particularly when the sample size is small and the population standard deviations are unknown. It is essential in situations where using a normal distribution is not appropriate due to small samples. The t distribution is similar to the normal distribution but has heavier tails, allowing for more variability. This extra variability accounts for the increased uncertainty associated with smaller samples. The form and shape of the t distribution depend on the degrees of freedom, which influence the steepness and tails of the distribution, making it more or less similar to a normal distribution.
Applications of the Student's t distribution include comparing means from two different groups to determine if they are statistically different from each other. This is particularly useful in real-world research scenarios such as medical trials, quality testing, and psychological studies, where often only small samples can be collected.

Degrees of freedom

Degrees of freedom (d.f.) are key in statistical calculations because they describe the number of values in the final calculation of a statistic that are free to vary. For example, when estimating a population mean from a sample, the degrees of freedom are directly related to the sample size. In essence, the greater the degrees of freedom, the more data you have, and the more reliable your statistical estimates will be. This measure becomes crucial when using statistical tools such as the Student's t distribution.
In the context of the Satterthwaite approximation in the exercise above, degrees of freedom are calculated to fine-tune the comparison of two sample means with differing variances. Typically, degrees of freedom are taken as the smaller of two samples minus one, but with the Satterthwaite formula, you achieve a more accurate estimation. This formula accounts for the variability within each sample, offering a more precise measure especially when the sample sizes are unequal.

Normal distribution

The normal distribution, also known as the Gaussian distribution, is a fundamental concept in statistics. It describes how the values of a variable are distributed. In the case of a normal distribution, most of the observations cluster around the central peak, and probabilities for values taper off equally toward both sides of the average or mean value. The classic shape is known as the 'bell curve,' and it's symmetric about the mean. This mean, along with the standard deviation, defines the distribution in terms of spread and center.
Many statistical tests, including t tests, are predicated on data being normally distributed or approximately so. However, in real-life applications, normality is not always given; thus tests such as those involving t distributions are used to mitigate non-normality issues. For smaller datasets, it's essential to make sure that the data fits such approximations, and if not, corrections and adjustments, such as the one described by Satterthwaite's formula, are to be considered.

Independent random samples

Independent random samples are a key assumption and concept in many statistical methods including the ones discussed here. An independent random sample means that the selection of a sample from a population is done in such a way that each individual has an equal and random chance of being selected, and one's selection does not influence others'.
This concept is crucial because independence ensures that any inferences made from the analysis hold valid and are not biased. It is especially important in hypothesis testing, regression analysis, and comparative studies. The exercise touches on this through the use of independent samples to apply the t distribution accurately. When samples are not independent, results can be skewed, resulting in inaccurate inferences and conclusions that might not be applicable beyond the analyzed data.