Question

The bill lengths of a population of male Blue Jays follow approximately a normal distribution with mean equal to \(25.4 \mathrm{~mm}\) and standard deviation equal to \(0.8 \mathrm{~mm}\) (as in Example 4.1.2). Find the 95 th percentile of the bill length distribution.

Short answer

The 95th percentile of the bill length for male Blue Jays is approximately \(25.4 \text{ mm} + (1.645 \times 0.8 \text{ mm}) = 26.716 \text{ mm} \).

Step-by-step solution

Understand the problem

The goal is to find the bill length for male Blue Jays that is greater than or equal to 95% of the population. We are given the mean bill length (\( \bar{x} = 25.4 \text{ mm} \) and the standard deviation (\( \text{s} = 0.8 \text{ mm} \) of a normally distributed population.

Use the z-score for the 95th percentile

To find the 95th percentile, we need to determine the z-score that corresponds to 95% of the area under the standard normal distribution. You can find this z-score using a standard normal distribution table or calculator. For the 95th percentile, the z-score is approximately 1.645.

Apply the z-score formula

Use the z-score formula to find the bill length corresponding to the 95th percentile. \( X = \bar{x} + (z \times \text{s}) \) where X is the 95th percentile bill length, \( \bar{x} \) is the mean, z is the z-score, and s is the standard deviation.

Calculate the 95th percentile bill length

Insert the values into the formula to obtain the 95th percentile bill length: \( X = 25.4 \text{ mm} + (1.645 \times 0.8 \text{ mm}) \) . Calculate the product and then add to the mean to find X.

Key concepts

Understanding Normal Distribution

The normal distribution, often known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In a normal distribution, the mean, median, and mode are all equal.

Here's an easy way to picture it: imagine a bell curve where the peak represents the average value, and the spread of the curve is determined by the standard deviation. When it comes to bird measurements like the bill lengths of Blue Jays, these are often normally distributed, which means most of the birds will have a bill length close to the mean, with fewer birds having very short or very long bills.

Mathematically, the formula for the normal distribution probability density function is quite complex, involving exponential functions. But for our purposes, we focus more on understanding its properties and how to use the distribution's table or software to find probabilities and percentiles, such as the 95th percentile in the given exercise.

Deciphering the Z-Score

A z-score, also known as a standard score, indicates how many standard deviations an element is from the mean. A key point when working with z-scores is they allow for comparison between different sets of data.

In the exercise, the z-score is used to find the 95th percentile. This score tells us the relative position of the bill length within the normal distribution of all Blue Jays' bill lengths. A z-score of 1.645 means that the 95th percentile of bill lengths is 1.645 standard deviations above the mean bill length.

Using z-scores can transform a normal distribution into a standard normal distribution, which has a mean of 0 and a standard deviation of 1. This standardization process simplifies the process of finding probabilities and percentiles because you can use standard normal distribution tables (or software) universally.

The Role of Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

This statistic is crucial because it gives context to the mean. Returning to the Blue Jays, a standard deviation of 0.8 mm signals that most bill lengths fall within a range that's relatively narrow around the average of 25.4 mm. Standard deviation helps in calculating the z-score and thus determines how far from the mean a certain percentile lies.

Exploring Probability Distributions

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. It tells us how likely it is for a random variable to take on a given value.

There are different types of probability distributions for different types of data, but they all share the common goal of helping to understand and predict the likelihood of various outcomes. The normal distribution is one type of probability distribution, and it's particularly useful because many phenomena in nature, including biological measurements like bill lengths, follow a pattern that resembles the bell curve of a normal distribution.

In the context of our exercise, understanding the normal distribution is essential to determine the 95th percentile, because we use the characteristics of this distribution to make inferences about the population of Blue Jays' bill lengths.