Problem 42 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The normal daily minimum temperature in degrees Fahrenheit for the months of January through December in San Francisco follows: $$\begin{array}{l}\begin{array}{l} 0\end{array}\\\\\begin{array}{llllll} 46.2 & 48.4 & 48.6 & 49.2 & 50.7 & 52.5 \\ 53.1 & 54.2 & 55.8 & 54.8 & 51.5 & 47.2 \end{array}\end{array}$$ Find the average and the median daily minimum temperature in San Francisco for these months.

Step-by-Step Solution

Answer

The average daily minimum temperature in San Francisco for the given months is 46.85°F, and the median daily minimum temperature is 51.1°F.

Step 1: Calculate the sum of the temperatures

First, let's calculate the sum of the given temperatures: \(46.2 + 48.4 + 48.6 + 49.2 + 50.7 + 52.5 + 53.1 + 54.2 + 55.8 + 54.8 + 51.5 + 47.2 = 562.2\)

Step 2: Calculate the average temperature

To find the average temperature, we need to divide the sum of the temperatures by the number of data points. Since we have 12 data points (months), we will divide 562.2 by 12: \(\text{Average temperature} = \frac{562.2}{12} = 46.85\) The average daily minimum temperature in San Francisco for these months is 46.85°F.

Step 3: Sort the temperatures in ascending order

Now, let's find the median daily minimum temperature. To do this, first, we need to sort the temperatures in ascending order: \(46.2, 47.2, 48.4, 48.6, 49.2, 50.7, 51.5, 52.5, 53.1, 54.2, 54.8, 55.8\)

Step 4: Find the median temperature

Since we have 12 data points, we need to find the average of the two middle values. In our sorted list, the two middle values are 50.7 and 51.5: \(\text{Median temperature} = \frac{50.7 + 51.5}{2} = \frac{102.2}{2} = 51.1\) The median daily minimum temperature in San Francisco for these months is 51.1°F. #Conclusion#: The average daily minimum temperature in San Francisco for the given months is 46.85°F, and the median daily minimum temperature is 51.1°F.

Key Concepts

Average Calculation

Calculating the average, also known as the mean, is a fundamental aspect of descriptive statistics and is a measure of central tendency. It showcases a typical value by dividing the sum of all numbers in a dataset by the quantity of numbers.

To calculate the average from a set of numerical data, you simply add up all the numbers and then divide by the number of data points you have. The average daily minimum temperature in our exercise is a simple arithmetic mean, which we found by summing all the monthly temperatures and dividing by 12, since there are 12 months.

The formula used is \[ \text{Average} = \frac{\sum{\text{data points}}}{\text{number of data points}} \] where \(\sum\) represents the summation of all the values. In practical situations, the average is useful to understand the overall trend of data, but it's important to note that it can be affected by extreme values, known as outliers.

Median Calculation

Unlike the average, the median is the middle value in a list of numbers that has been sorted in ascending or descending order. If there's an even number of observations, the median is the average of the two middle numbers.

In the exercise, with 12 months of temperature data, we first sorted the values in order before calculating the median. With an even number of data points, we took the average of the two middle values, which is the sixth and seventh values when sorted. The median, hence, shows the central value of our dataset and is particularly useful in datasets with outliers as it's not skewed by extremely high or low values.

The general formula for the median in an ordered set with an even number of terms is given by \[ \text{Median} = \frac{\text{n/2-th term} + \text{(n/2+1)-th term}}{2} \] and this measure is a better indicator of the 'middle' of a dataset, especially when the dataset includes outliers.

Data Sorting

Data sorting is a crucial step in many mathematical problem-solving processes, especially when determining the median of a dataset. Sorting is the arrangement of data into a meaningful order to simplify analysis.

In our case, we arranged the monthly temperatures in ascending order to easily locate the median. Sorting is not just beneficial for finding the median; it helps in identifying patterns and making comparisons. It also lays the foundation for other statistical analyses, such as quartiles and percentiles calculation.

To sort data efficiently, numerous algorithms can be applied, each with its advantages based on the size and nature of the data set. For small datasets or preliminary analysis, manual sorting or basic computer programs are usually sufficient.

Mathematical Problem-Solving

Mathematical problem-solving encompasses a structured approach towards tackling math-related questions. Our problem of finding the average and median temperatures showcases this structured method: starting with an understanding of the problem, followed by data organization, calculation, and interpretation of results.

In any mathematical problem-solving scenario, it's vital to:

• Clearly identify what you're asked to find.
• Gather all necessary information and organize it logically.
• Select and apply appropriate calculations.
• Review the solution to ensure it is reasonable and accurately answers the question.

The steps taken in our exercise demonstrate this process, where sorting, averaging, and median calculation all play a role. Correct application of these methods leads to effective problem resolution and a deeper understanding of the concepts at hand.

Each step performed falls into a broad category of mathematical problem-solving, which is a critical skill for students to master as they advance through their education.