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Chapter 3: Problem 21

Poe, the Clydesdale horse has a world record breaking height of 20.2 hands. All Clydesdale horses have a mean height of 16.5 hands and a standard deviation of 1.85 hands. The last Great Dane to hold the world record for dog height was Gibson who was \(107 \mathrm{~cm}\) tall. Great Danes have a mean height of \(81 \mathrm{~cm}\) and a standard deviation of \(13 \mathrm{~cm}\). Which animal is taller compared to their respective breed? Explain your answer using Z-scores.

Short Answer

Expert verified
Poe and Gibson are equally tall compared to their breeds, with Z-scores of 2.00.

Step by step solution

01

Understanding Z-score formula

The Z-score indicates how many standard deviations an element is from the mean. It is calculated using the formula: \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
02

Calculate Z-score for Poe

For Poe, the Clydesdale horse, \( X = 20.2 \) hands, \( \mu = 16.5 \) hands, and \( \sigma = 1.85 \) hands. Substitute into the Z-score formula: \[ Z_{\text{Poe}} = \frac{20.2 - 16.5}{1.85} \] \[ Z_{\text{Poe}} = \frac{3.7}{1.85} = 2.00 \].
03

Calculate Z-score for Gibson

For Gibson, the Great Dane, \( X = 107 \) cm, \( \mu = 81 \) cm, and \( \sigma = 13 \) cm. Substitute into the Z-score formula: \[ Z_{\text{Gibson}} = \frac{107 - 81}{13} \] \[ Z_{\text{Gibson}} = \frac{26}{13} = 2.00 \].
04

Compare Z-scores

Both Poe and Gibson have a Z-score of 2.00. This means that each is exactly 2 standard deviations above the average height for their respective breeds.
05

Conclusion

Since both animals have the same Z-score, they are equally remarkable in terms of height compared to their breeds. Thus, neither is taller compared to their breed than the other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Clydesdale horse height
Clydesdale horses are known for their impressive stature and are a popular breed often seen in parades and used for heavy farm work. On average, Clydesdale horses have a mean height of 16.5 hands. This average height serves as a benchmark to compare other Clydesdales to.

Poe, a Clydesdale horse that reached a record-breaking height of 20.2 hands, stands out significantly when compared to this average. To understand how extraordinary Poe's height is, we use a standard measure called the Z-score. It tells us how many standard deviations Poe's height is from the average height of Clydesdales.
  • Poe's height: 20.2 hands
  • Average (mean) Clydesdale height: 16.5 hands
  • Standard deviation: 1.85 hands
This comparison helps us see just how exceptional Poe's height is compared to his breed!
Great Dane height
Great Danes are one of the largest dog breeds, known for their towering presence. The average height for a Great Dane is typically around 81 cm. Gibson, a former record-holding Great Dane, stood at a whopping 107 cm tall. This exceptional height makes Gibson quite remarkable when compared to the average height of his breed.

The difference between Gibson's height and the mean height of Great Danes can also be assessed using the Z-score method, which quantifies how far Gibson's height deviates from the norm.
  • Gibson's height: 107 cm
  • Average (mean) Great Dane height: 81 cm
  • Standard deviation: 13 cm
By using these figures, Gibson's remarkable stature relative to his peers can be appreciated.
standard deviation
Standard deviation is a crucial concept in statistics that indicates how spread out the numbers are in a data set. A smaller standard deviation means that the data points tend to be close to the mean, whereas a larger standard deviation indicates a wide spread.

For the Clydesdale horses, the standard deviation of 1.85 hands shows the amount of variation or dispersion from the mean height. In contrast, the Great Danes have a standard deviation of 13 cm, highlighting more variability in terms of height for this breed.
  • Clydesdales: 1.85 hands
  • Great Danes: 13 cm
Understanding standard deviation is key when analyzing how individual examples, like Poe and Gibson, compare to the average height of their respective breeds.
mean height comparison
The mean, or average height, provides a central value around which other measurements can be compared. It serves as a starting point for comparing how different individuals measure up within a population. This is where the Z-score comes in handy. By computing a Z-score, we gauge how far a particular measurement deviates from the average.

In comparing mean heights:
  • Clydesdales have an average height of 16.5 hands.
  • Great Danes have an average height of 81 cm.
We can use these averages to calculate how extraordinary Poe's height of 20.2 hands and Gibson's height of 107 cm are relative to their respective breeds. Each has a Z-score of 2.00, instantly indicating both are exactly two standard deviations beyond the mean. So, in terms of deviation from the mean, both animals stand out equally within their groups!

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