Problem 14 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A panel of 64 economists was asked to predict the average unemployment rate for the upcoming year. The results of the survey follow: $$\begin{array}{lccccccc} \hline \text { Unemployment } & & & & & & & \\ \text { Rate, \% } & 4.5 & 4.6 & 4.7 & 4.8 & 4.9 & 5.0 & 5.1 \\ \hline \text { Economists } & 2 & 4 & 8 & 20 & 14 & 12 & 4 \\ \hline\end{array}$$ Based on this survey, what does the panel expect the average unemployment rate to be next year?

Step-by-Step Solution

Answer

Based on the panel's predictions, the expected average unemployment rate for the upcoming year is approximately \(4.8375\%\).

Step 1: Set up the weighted mean formula

To compute the expected unemployment rate, we'll use the weighted mean formula, which is given by: $$Expected \, Unemployment \, Rate = \frac{\sum_{i=1}^{n}(w_i \cdot x_i)}{\sum_{i=1}^{n}w_i}$$ where \(w_i\) represents the weight (number of economists) for each unemployment rate prediction \(x_i\) and \(n\) is the total number of different unemployment rates predicted.

Step 2: Solve for the expected unemployment rate

Using the data from the table, we have: Weighted Sum = \((2 \times 4.5) + (4 \times 4.6) + (8 \times 4.7) + (20 \times 4.8) + (14 \times 4.9) + (12 \times 5.0) + (4 \times 5.1)\) Total Weight = \(2 + 4 + 8 + 20 + 14 + 12 + 4\) Now, substitute these values into the weighted mean formula: \(Expected \, Unemployment \, Rate = \frac{Weighted \, Sum}{Total \, Weight}\) \(Expected \, Unemployment \, Rate = \frac{(2 \times 4.5) + (4 \times 4.6) + (8 \times 4.7) + (20 \times 4.8) + (14 \times 4.9) + (12 \times 5.0) + (4 \times 5.1)}{2 + 4 + 8 + 20 + 14 + 12 + 4}\) \(Expected \, Unemployment \, Rate = \frac{9 + 18.4 + 37.6 + 96 + 68.6 + 60 + 20.4}{64}\) \(Expected \, Unemployment \, Rate = \frac{309.6}{64}\) \(Expected \, Unemployment \, Rate ≈ 4.8375\)

Step 3: Interpret the result

Based on the panel's predictions, the expected average unemployment rate for the upcoming year is approximately 4.8375%.

Key Concepts

Expected Unemployment Rate

The expected unemployment rate is crucial for assessing the future health of an economy. It’s a statistical probability that estimates the level of joblessness within a country for a specified period. Economists predict this rate based on various indicators, including economic policy, market trends, and historical data.

In our exercise, a panel of economists provided their predictions for the unemployment rate for the upcoming year. To find a single figure that best represents the panel's collective forecast, we used a weighted average. This allowed us to consider not just the different rates, but how many economists backed each prediction.

Weighted Average

A weighted average is a mean where different values in the dataset contribute unequally to the final average. This concept is fundamental when individual values carry different significance or 'weight'.

In determining the expected unemployment rate, each economist's prediction was given a weight equivalent to the number of economists who agreed with that particular rate. The weighted average is more representative than a simple average when contributions to the dataset vary in importance or frequency. It amplifies the influence of more commonly held beliefs while still considering minority opinions.

Economic Predictions

Economic predictions, like the expected unemployment rate, are forecasts that guide governments, businesses, and individuals in decision-making. They denote what economists believe will happen in the economic landscape based on current conditions and models.

It's essential to gather a range of estimates, as predictions can vary based on the methodology and perspective of different economists. A weighted mean calculation incorporates the diversity of these views and yields a number that is more accurate than any single expert's forecast. It's a powerful tool for turning a collection of expert opinions into actionable knowledge for planning and policy.