Key Concepts and Formulas

• Mean ($\bar{X}$): For a set of $n$ observations $x_1, x_2, \ldots, x_n$, the mean is the sum of all observations divided by the number of observations. $\bar{X} = \frac{\sum_{i=1}^n x_i}{n}$ This implies that the sum of observations can be found as $\sum_{i=1}^n x_i = n \times \bar{X}$.

• Variance ($\sigma^2$): Variance measures the spread of data points from their mean. A computationally convenient formula for variance is: $\sigma^2 = \frac{\sum_{i=1}^n x_i^2}{n} - (\bar{X})^2$ This formula can be rearranged to find the sum of squares of observations: $\sum_{i=1}^n x_i^2 = n \times (\sigma^2 + (\bar{X})^2)$.

Step-by-Step Solution

Let the five observations be $x_1, x_2, x_3, x_4, x_5$. We are given information about these five observations and also about the first four observations ($x_1, x_2, x_3, x_4$).

• Given for five observations ($n_5=5$):

  • Mean ($\bar{X}_5$) = $\frac{24}{5}$
  • Variance ($\sigma_5^2$) = $\frac{194}{25}$

• Given for the first four observations ($n_4=4$):

  • Mean ($\bar{X}_4$) = $\frac{7}{2}$

Our goal is to find the variance of these first four observations ($\sigma_4^2$).

Step 1: Calculate the Sum and Sum of Squares for the Five Observations

• What and Why: We first use the given mean and variance of all five observations to determine their total sum and the total sum of their squares. These values provide a complete picture of the initial dataset, which will then allow us to isolate the properties of the subset of four observations.

• 1.1. Calculate the sum of the five observations ($\sum_{i=1}^5 x_i$): Using the mean formula: $\sum_{i=1}^5 x_i = n_5 \times \bar{X}_5 = 5 \times \frac{24}{5} = 24$

• 1.2. Calculate the sum of squares of the five observations ($\sum_{i=1}^5 x_i^2$): Using the variance formula: $\sigma_5^2 = \frac{\sum_{i=1}^5 x_i^2}{n_5} - (\bar{X}_5)^2$ Substitute the given values: $\frac{194}{25} = \frac{\sum_{i=1}^5 x_i^2}{5} - \left(\frac{24}{5}\right)^2$ $\frac{194}{25} = \frac{\sum_{i=1}^5 x_i^2}{5} - \frac{576}{25}$ Rearranging the equation to solve for $\frac{\sum_{i=1}^5 x_i^2}{5}$: $\frac{\sum_{i=1}^5 x_i^2}{5} = \frac{194}{25} + \frac{576}{25} = \frac{194 + 576}{25} = \frac{770}{25}$ Now, multiply by 5 to get the sum of squares: $\sum_{i=1}^5 x_i^2 = 5 \times \frac{770}{25} = \frac{770}{5} = 154$

Step 2: Calculate the Sum of the First Four Observations

• What and Why: We use the given mean of the first four observations to find their sum ($\sum_{i=1}^4 x_i$). This sum is directly needed for calculating their variance and also helps in determining the value of the fifth observation ($x_5$).

• Calculation of $\sum_{i=1}^4 x_i$: Using the mean formula for $n_4=4$: $\sum_{i=1}^4 x_i = n_4 \times \bar{X}_4 = 4 \times \frac{7}{2} = 14$

Step 3: Determine the Value of the Fifth Observation ($x_5$)

• What and Why: The fifth observation ($x_5$) is the link between the complete set of five observations and the subset of four. By subtracting the sum of the first four observations from the total sum of all five observations, we can find the individual value of $x_5$. This value is essential for isolating the sum of squares of the first four observations.

• Calculation of $x_5$: We know that the sum of five observations is the sum of the first four plus the fifth: $\sum_{i=1}^5 x_i = \left(\sum_{i=1}^4 x_i\right) + x_5$ Substitute the sums calculated in Step 1.1 and Step 2: $24 = 14 + x_5$ $x_5 = 24 - 14 = 10$

Step 4: Calculate the Sum of Squares for the First Four Observations ($\sum_{i=1}^4 x_i^2$)

• What and Why: To compute the variance of the first four observations, we need their sum of squares. We can obtain this by subtracting the square of the fifth observation ($x_5^2$) from the total sum of squares of all five observations.

• Calculation of $\sum_{i=1}^4 x_i^2$: We know that: $\sum_{i=1}^5 x_i^2 = \left(\sum_{i=1}^4 x_i^2\right) + x_5^2$ From Step 1.2, $\sum_{i=1}^5 x_i^2 = 154$. From Step 3, $x_5 = 10$, so $x_5^2 = 10^2 = 100$. Substitute these values: $154 = \left(\sum_{i=1}^4 x_i^2\right) + 100$ $\sum_{i=1}^4 x_i^2 = 154 - 100 = 54$

Step 5: Calculate the Variance of the First Four Observations

• What and Why: We now have all the necessary components for the first four observations: the number of observations ($n_4=4$), their mean ($\bar{X}_4=\frac{7}{2}$), and their sum of squares ($\sum_{i=1}^4 x_i^2=54$). We can directly apply the variance formula to calculate the final answer.

• Calculation of $\sigma_4^2$: Using the variance formula for $n_4=4$: $\sigma_4^2 = \frac{\sum_{i=1}^4 x_i^2}{n_4} - (\bar{X}_4)^2$ Substitute the values: $\sigma_4^2 = \frac{54}{4} - \left(\frac{7}{2}\right)^2$ $\sigma_4^2 = \frac{54}{4} - \frac{49}{4}$ $\sigma_4^2 = \frac{54 - 49}{4} = \frac{5}{4}$

Common Mistakes & Tips

• Distinguish 'n': Always be mindful of the number of observations ($n$) you are working with at each stage. The problem involves switching between $n=5$ and $n=4$, which is a common point of error.
• Formula Recall and Rearrangement: Ensure you have a strong grasp of the mean and variance formulas and can confidently rearrange them to solve for sums or sums of squares as needed.
• Arithmetic Precision: Be meticulous with all calculations, especially those involving fractions and squaring terms. A small arithmetic mistake can lead to an incorrect final answer.

Summary

This problem is a classic application of statistical definitions, demonstrating how to relate the properties of a complete dataset to those of its subset. The strategy involved systematically deriving the sum and sum of squares for the initial set of five observations, using the mean of the first four observations to find their sum, and then determining the value of the unique fifth observation. By subtracting the contribution of this fifth observation from the totals, we found the sum of squares for the first four observations. Finally, applying the variance formula to this subset yielded the desired result. The variance of the first four observations is $\frac{5}{4}$.

Final Answer

The final answer is $\boxed{\frac{5}{4}}$, which corresponds to option (A).