Question

Suppose a certain population of observations is normally distributed. What percentage of the observations in the population (a) are within ±1.5 standard deviations of the mean? (b) are more than 2.5 standard deviations above the mean? (c) are more than 3.5 standard deviations away from (above or below) the mean?

Short answer

The percentage of observations (a) within ±1.5 standard deviations of the mean is approximately 86.64%, (b) more than 2.5 standard deviations above the mean is approximately 0.62%, and (c) more than 3.5 standard deviations away from the mean is approximately 0.05%.

Step-by-step solution

Understanding the Problem

We are given a normally distributed population and asked to find the proportion of observations within certain ranges based on standard deviations from the mean. (a) requires us to find the percentage within ±1.5 standard deviations, (b) asks for the percentage more than 2.5 standard deviations above the mean, and (c) wants the percentage more than 3.5 standard deviations away from the mean.

Using the Standard Normal Distribution

To find these percentages, we use the standard normal distribution (Z-distribution), where the mean is 0 and the standard deviation is 1. We can use a Z-table or standard normal distribution calculator to find the probabilities corresponding to certain Z-scores.

Finding the Z-scores

Calculate the Z-scores for ±1.5, +2.5, and ±3.5 standard deviations from the mean, using the formula Z = (X - mean) / standard deviation. Since the mean is 0 for the Z-distribution: Z = ±1.5 for (a), Z = +2.5 for (b), and Z = ±3.5 for (c).

Calculating the Percentage for (a)

Look up the Z-scores (±1.5) in the Z-table or use a calculator to find the cumulative probabilities. The percentage within ±1.5 standard deviations is the probability of being less than 1.5 standard deviations above the mean minus the probability of being less than 1.5 standard deviations below the mean.

Calculating the Percentage for (b)

Look up the Z-score (+2.5) to find the cumulative probability of being less than 2.5 standard deviations above the mean. Subtract this probability from 1 (or 100%) to find the percentage that is more than 2.5 standard deviations above the mean.

Calculating the Percentage for (c)

Look up the Z-score (±3.5) and find the cumulative probability for being less than 3.5 standard deviations above the mean and for being more than 3.5 standard deviations below the mean (which is 1 minus the cumulative probability for -3.5). Add these two probabilities together to get the percentage more than 3.5 standard deviations away from the mean.

Key concepts

Standard Deviation

Understanding standard deviation is essential when you're working with normally distributed data. A standard deviation is a measurement of the amount of variation or dispersion in a set of values. In simpler terms, it tells us how spread out the numbers are from the mean (average) of the data set.

For a normal distribution, approximately 68% of the data falls within one standard deviation from the mean, roughly 95% falls within two standard deviations, and about 99.7% lie within three standard deviations. This rule is known as the Empirical Rule or the 68-95-99.7 rule.

In the context of the textbook exercise, understanding standard deviation allows us to characterize intervals around the mean and to calculate the proportion of observations that would lie in these intervals. It's important to note that a higher standard deviation indicates greater variability in the data set. Conversely, a smaller standard deviation suggests that the observations are closer to the mean.

Z-score

The Z-score is another fundamental concept in statistics, particularly when you're dealing with normal distributions. It is a measure of how many standard deviations an individual data point is from the mean of the data set. You calculate a Z-score using the formula: \( Z = \frac{X - \text{mean}}{\text{standard deviation}} \), where \(X\) is a single observation.

This standardized score is immensely useful as it allows you to compare data points from different normal distributions, regardless of their respective means or standard deviations.

Remember, a Z-score of 0 corresponds to the mean, while positive or negative Z-scores indicate positions above or below the mean. In our exercise, we calculated Z-scores to transform our observations into a common scale that can be easily referred to the standard normal distribution table (or Z-table), which then allows us to find cumulative probabilities.

Cumulative Probability

Finally, cumulative probability is a pivotal term in understanding normal distributions. It represents the probability that a random variable is less than or equal to a particular value. In terms of a normal distribution curve, it's the area under the curve to the left of a given Z-score.

For instance, if we wish to determine the cumulative probability for a Z-score of 1.5, we are essentially looking for the area under the curve from the leftmost end up to the point that corresponds to 1.5 standard deviations above the mean. Cumulative probabilities are what the Z-table provides or can be calculated using a standard normal distribution calculator.

In summary, cumulative probabilities help us to quantify the likelihood of observing a value within certain bounds of a normal distribution, which directly informed our ability to solve parts (a), (b), and (c) of our exercise.