Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.


Water Footprint A water footprint is a measure of the appropriation of fresh water. The water footprint (in cubic meters) for a kilogram of wheat can be approximated by a normal distribution, as shown in the figure. (Source: Ecological Indicators)





b. What water footprint represents the 29th percentile?

Verified step by step guidance

1. Step 1: Understand the problem. The question asks for the water footprint value that corresponds to the 29th percentile in a normal distribution. The normal distribution is defined by a mean (μ = 1.34 m³/kg) and a standard deviation (σ = 0.80 m³/kg). Percentiles represent the position of a value in the distribution relative to the rest of the data.
2. Step 2: Convert the percentile into a z-score. The z-score is a standardized value that corresponds to the given percentile in the standard normal distribution. Use a z-score table or statistical software to find the z-score for the 29th percentile. For example, the z-score for the 29th percentile is approximately -0.55.
3. Step 3: Use the z-score formula to find the corresponding value in the normal distribution. The formula is: x = μ + zσ, where x is the value in the normal distribution, μ is the mean, z is the z-score, and σ is the standard deviation.
4. Step 4: Substitute the values into the formula. Plug in μ = 1.34, z = -0.55, and σ = 0.80 into the formula x = μ + zσ.
5. Step 5: Simplify the expression to find the water footprint value corresponding to the 29th percentile. This will give you the final result in cubic meters.

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean (μ) and standard deviation (σ). In this context, the water footprint of wheat follows a normal distribution, indicating that most values cluster around the mean, with fewer occurrences as you move away from it.

Percentiles

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, the 29th percentile represents the value below which 29% of the data points lie. Understanding percentiles is crucial for interpreting the distribution of data and identifying specific thresholds.

Z-scores

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It is calculated by subtracting the mean from the value and dividing by the standard deviation. Z-scores are essential for finding percentiles in a normal distribution, as they allow for the conversion of raw scores into a standardized format.