Problem 4 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

The sample mean and sample median of the uncorrected Wick temperature data (in degrees Fahrenheit) are \(44.7\) and 42 . We transform the data from degrees Fahrenheit \(\left(x_{i}\right)\) to degrees Celsius \(\left(y_{i}\right)\) by means of the formula $$ y_{i}=\frac{5}{9}\left(x_{i}-32\right), $$ which gives the following dataset $$ \begin{array}{lllllllllll} \frac{55}{9} & \frac{55}{9} & 5 & 5 & 5 & \frac{50}{9} & \frac{55}{9} & \frac{130}{9} & \frac{130}{9} & 5 & 5 . \end{array} $$ a. Check that \(\bar{y}_{n}=\frac{5}{9}\left(\bar{x}_{n}-32\right)\). b. Is it also true that \(\operatorname{Med}\left(y_{1}, \ldots, y_{n}\right)=\frac{5}{9}\left(\operatorname{Med}\left(x_{1}, \ldots, x_{n}\right)-32\right)\) ? c. Suppose we have a dataset \(x_{1}, x_{2}, \ldots, x_{n}\) and construct \(y_{1}, y_{2}, \ldots, y_{n}\) where \(y_{i}=a x_{i}+b\) with \(a\) and \(b\) being real numbers. Do similar relations hold for the sample mean and sample median? If so, state them.

Step-by-Step Solution

Answer

a. Verified: \( \bar{y}_{n} = 7.0556 \). b. False, \( \operatorname{Med}(y) \) is not \( 5.5556 \). c. Holds true for means, medians transform with \( a > 0 \).

Step 1: Verify the transformation for the mean

To verify \( \bar{y}_{n} = \frac{5}{9} (\bar{x}_{n} - 32) \), we take the given sample mean \( \bar{x}_{n} = 44.7 \). Using the transformation formula, we calculate:\[\bar{y}_{n} = \frac{5}{9} (\bar{x}_{n} - 32) = \frac{5}{9} (44.7 - 32) = \frac{5}{9} \times 12.7 = \frac{63.5}{9} \].Computing this gives \( \bar{y}_{n} = 7.0556 \).

Step 2: Calculate and verify transformed median

For the sample median transformation, we take \( \operatorname{Med}(x_1, \ldots, x_n) = 42 \). Applying the transformation, we have:\[\operatorname{Med}(y_1, \ldots, y_n) = \frac{5}{9} (\operatorname{Med}(x_1, \ldots, x_n) - 32) = \frac{5}{9} \times (42 - 32) = \frac{5}{9} \times 10 = \frac{50}{9} = 5.5556\],which is not the same as directly calculating the median of the \( y_i \) dataset, which is \( 5 \). Thus, the medians do not transform directly in this manner.

Step 3: Generalize for transformations y_i = ax_i + b

For a general transformation \( y_i = ax_i + b \), the transformed sample mean is similarly given by:\[ \bar{y}_n = a\bar{x}_n + b \],which holds true based on linearity. However, for the sample median, \( \operatorname{Med}(y_1, \ldots, y_n) = a \times \operatorname{Med}(x_1, \ldots, x_n) + b \) only if \( a > 0 \). This holds because the ordering of medians is preserved under positive scaling and translation transformations.

Key Concepts

Understanding the Sample Mean

The sample mean is a powerful tool in statistics for summarizing the central tendency of a dataset. It is calculated by adding up all the individual data points and then dividing by the number of data points. This provides a numerical average that represents the dataset. For example, consider a set of temperatures recorded in degrees Fahrenheit. In the case of our exercise, the sample mean, \( \bar{x}_n \), of the initial temperature data is found to be 44.7 degrees Fahrenheit. This number gives us a sense of the average temperature in the dataset.

To convert the sample mean from Fahrenheit to Celsius, we can use the given transformation formula: \( \bar{y}_n = \frac{5}{9} (\bar{x}_n - 32) \). This transformation adjusts the scale of the temperature from Fahrenheit to Celsius, which can be useful for different scientific analyses or regulatory requirements. After using this formula, the calculated mean temperature in Celsius, \( \bar{y}_n \), becomes 7.0556.

By understanding the concept of sample mean and the ability to transform it into different units, students gain insight into how data can be summarized and reinterpreted through different perspectives.

Exploring the Sample Median

Unlike the mean, the sample median is the middle value of a dataset when it is ordered from smallest to largest. It divides the dataset into two equal halves, making it a robust measure of central tendency especially in the presence of outliers. In our exercise, the sample median of the original temperature data is 42 degrees Fahrenheit. This tells us that half of the values in the dataset are below 42, and half are above.

When we attempt to transform the sample median to Celsius using \( \frac{5}{9} (\operatorname{Med}(x_1, \ldots, x_n) - 32) \), we find that the transformed median \( \operatorname{Med}(y_1, \ldots, y_n) \) becomes 5.5556. However, upon calculating the median directly from the Celsius-transformed dataset, it turns out to be 5. This shows that unlike the mean, the mathematical transformation of the median does not yield the same value as the transformed dataset's actual median.

The discrepancy arises because the median is a positional statistic rather than a summed value. Understanding this concept allows students to discern situations where direct transformations might not hold true for median calculations.

The Role of Data Transformation

Data transformation is a process used to convert data from one format or unit of measurement to another. It is often applied to improve data interpretability or compliance with analysis prerequisites. In the context of our exercise, the transformation process involves converting temperature data from degrees Fahrenheit to degrees Celsius.

For a general transformation of the form \( y_i = ax_i + b \), the sample mean of the transformed data \( \bar{y}_n = a\bar{x}_n + b \) maintains the property of linearity, meaning it transforms in a straightforward manner under linear transformation. However, for the sample median, the transformation \( \operatorname{Med}(y_1, \ldots, y_n) = a \times \operatorname{Med}(x_1, \ldots, x_n) + b \) holds only if \( a > 0 \). This is due to the fact that scaling and translating the data maintain the order of the data points only when \( a \) is positive.

Through understanding data transformation, students can effectively apply it to various types of analysis, ensuring meaningful comparisons and conclusions across different units and scales.