Question

One method of arriving at economic forecasts is to use a consensus approach. A forecast is obtained from each of a large number of analysts, and the average of these individual forecasts is the consensus forecast. Suppose the individual 2008 January prime interest rate forecasts of economic analysts are approximately normally distributed with the mean equal to \(8.5 \%\) and a standard deviation equal to \(0.2 \% .\) If a single analyst is randomly selected from among this group, what is the probability that the analyst's forecast of the prime rate will take on these values? a. Exceed \(8.75 \%\) b. Be less than \(8.375 \%\)

Short answer

Answer: The probability that the analyst's forecast of the prime rate will exceed 8.75% is 10.56%, and the probability that the forecast will be less than 8.375% is 26.59%.

Step-by-step solution

Identifying mean, standard deviation and the given values

In this problem, the mean (μ) is \(8.5 \%\), the standard deviation (σ) is \(0.2 \%\), and our given values are \(8.75 \%\) and \(8.375 \%\) for parts a and b.

Calculate the z-scores

Z-scores tell us how many standard deviations away from the mean a particular value is. To calculate the z-score, use the formula: z = (X - μ) / σ For part a: X = \(8.75 \%\), μ = \(8.5 \%\), and σ = \(0.2 \%\) z = (\(8.75 - 8.5) / 0.2\) = \(0.25 /0.2\) = \(1.25\) For part b: X = \(8.375 \%\) z = (\(8.375 - 8.5) / 0.2\) = \(-0.125 /0.2\) = \(-0.625\)

Use the z-table to find the probabilities

Now that we have the z-scores for both parts a and b, we can use the z-table to find the respective probabilities. Refer to a standard normal table to find the probabilities corresponding to the calculated z-scores. For part a: P(Z > 1.25) = 1 - P(Z ≤ 1.25) = 1 - 0.8944 = 0.1056 For part b: P(Z ≤ -0.625) = 0.2659

Final answer

Now we have the probabilities for both parts a and b. a. The probability that the analyst's forecast of the prime rate will exceed \(8.75 \%\) is 0.1056, or \(10.56 \%\). b. The probability that the analyst's forecast of the prime rate will be less than \(8.375 \%\) is 0.2659, or \(26.59 \%\).

Key concepts

Normal distribution

The concept of a normal distribution is crucial in statistics and helps us understand data patterns. A normal distribution, often referred to as a Gaussian distribution, is a bell-shaped curve symmetrical around its mean. This symmetry implies that most of the data points cluster around the central peak and the likelihood of extreme deviations decreases as you move away from the mean. It's a continuous probability distribution that is very commonly used in various fields due to its important properties.

Key characteristics of a normal distribution include:

• The mean, median, and mode are all equal.
• The curve is symmetric at the center, meaning the left side of the curve is a mirror image of the right side.
• It follows the empirical rule, also known as the 68-95-99.7 rule, which tells us that:
  • About 68% of values lie within one standard deviation of the mean.
  • About 95% of values lie within two standard deviations.
  • About 99.7% of values lie within three standard deviations.

Understanding the normal distribution helps in computing probabilities and is foundational for hypothesis testing and creating models that predict outcomes.

Z-score

A Z-score is a statistical measure that describes a value's position in relation to the mean of a group of values, expressed in terms of standard deviations from the mean. Calculating a Z-score allows you to determine how unusual or typical a certain value is compared to other data points.

The formula for calculating a Z-score is:\( z = \frac{X - \mu}{\sigma} \)where

• \(X\) is the value for which you want the Z-score,
• \(\mu\) is the mean of the data,
• \(\sigma\) is the standard deviation.

A Z-score of 0 indicates the value is exactly average, while a positive Z-score indicates it's above average, and a negative Z-score indicates it's below average.

In the given exercise, Z-scores help to determine the probability of an analyst forecasting a particular prime interest rate. By converting individual forecasts to Z-scores, we make them comparable with a standard normal distribution, allowing for straightforward probability calculations using the Z-table.

Consensus forecast

The consensus forecast approach involves aggregating forecasts from a group of analysts or experts to derive a unified prediction. This method capitalizes on the diversity of opinions and analyses to produce more reliable and accurate forecasts than an individual forecast might provide.

A consensus forecast is obtained by calculating the average of all individual forecasts. The logic behind using a consensus is that errors from individual analysts may cancel each other out. Consequently, the group prediction tends to be more stable over time.

• Reduces individual biases, leading to a more balanced perspective.
• Draws from a wide range of methodologies and insights.
• Increases predictability and reduces the impact of any outlier predictions.

This gathered prediction is particularly useful in fields like economics, as seen in the original exercise, where diverse factors can make individual analysis complex and uncertain. The consensus forecast is a more weighted and dependable projection for decision-making.

Standard deviation

Standard deviation is a fundamental measure used in statistics to quantify the amount of variation or dispersion in a set of data values. Essentially, it tells us how spread out the values in a dataset are around the mean, providing insights into the variability of the data.

The formula for standard deviation \( \sigma \) is:\[ \sigma = \sqrt{\frac{\sum_{i=1}^{n} (X_i - \mu)^2}{n}} \]where

• \(X_i\) are the individual data points,
• \(\mu\) is the mean of the data,
• \(n\) is the number of data points.

A smaller standard deviation indicates that the data points tend to be close to the mean, whereas a larger standard deviation indicates more spread out data points.

In our exercise, the standard deviation of 0.2% indicates that most economists' interest rate forecasts are close to the mean of 8.5%, suggesting a high level of agreement among them. Understanding this concept is crucial for evaluating the reliability and risk associated with predictions like the consensus forecast in the problem.