Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Suppose \(X\) is a random variable with mean \(\mu\) and standard deviation \(\sigma\). If a large number of trials is observed, at least what percentage of these values is expected to lie between \(\mu-2 \sigma\) and \(\mu+2 \sigma ?\)

Step-by-Step Solution

Answer

At least \(95\%\) of the values are expected to lie between \(\mu - 2\sigma\) and \(\mu + 2\sigma\) after a large number of trials, based on the empirical rule for a normal distribution.

Step 1: Identify the interval

First, we identify the interval of which we want to know the expected percentage of values: The interval is given as μ - 2σ and μ + 2σ.

Step 2: Recall the Empirical Rule

Remember the empirical rule for a normal distribution: Approximately 68% of the values lie within 1 standard deviation from the mean, or μ ± σ Approximately 95% of the values lie within 2 standard deviations from the mean, or μ ± 2σ Approximately 99.7% of the values lie within 3 standard deviations from the mean, or μ ± 3σ

Step 3: Apply the Empirical Rule

Now, using the empirical rule, we look for the percentage of values expected to lie between μ - 2σ and μ + 2σ. The empirical rule states that approximately 95% of values lie within 2 standard deviations from the mean, or within the given interval. Hence, at least 95% of the values are expected to lie between μ - 2σ and μ + 2σ after a large number of trials.

Key Concepts

Normal Distribution

The concept of normal distribution is fundamental to understanding how data tends to spread around a central value, typically the mean. Often referred to as a 'bell curve,' a normal distribution represents how the values of a random variable are distributed in such a way that most observations cluster around the mean (central peak) and the probability of values further away from the mean tapers off symmetrically in both directions.

Under a normal distribution, specific percentages of data fall within certain distances from the mean measured in units of standard deviation. For students, visualizing this concept is critical. Imagine a perfectly symmetrical bell-shaped graph where the bulk of the data congregates in the middle and thins out towards the edges. This pattern is not just theoretical but occurs in many natural phenomena and human-related measurements like heights, test scores, and errors in measurements.

Standard Deviation

Standard deviation, often denoted as \(\sigma\), is a measure that quantifies the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points are close to the mean of the data set, whereas a high standard deviation indicates that the data points are spread out over a wider range of values.

To elaborate, if we were comparing the test scores of two classes where one class had a low standard deviation and the other a high one, the former would have scores concentrated around the class average while the latter would show scores that are more spread out. It's essential for students to grasp that standard deviation is key to understanding how 'spread out' data is in any given data set, which directly relates to the predictability and reliability of results. Understanding standard deviation in the context of the empirical rule allows one to make predictions about where most data points lie in relation to the mean.

Statistical Significance

Statistical significance is a concept that helps us determine if a result is not due to random chance but instead likely to be attributable to a specific cause. In essence, a statistically significant result is one that is unlikely to happen by chance. This concept is essential when trying to draw conclusions from data, especially in experiments and studies.

The determination of statistical significance is typically done through hypothesis testing, where a p-value (probability value) is calculated. If this p-value is lower than a predefined threshold, usually 0.05 or 5%, the result is considered statistically significant. However, students should be aware that statistical significance does not imply that a result is practically significant or that it has a large effect size. It merely indicates that there is statistical evidence to suggest that the observed effect is unlikely to be due to random variation.