Question

Miniature poodles are thought to have a mean height of 12 inches and a standard deviation of \(1.8\) inches. If height is measured to the nearest inch, find the percentage of poodles having a height exceeding 14 inches.

Short answer

Approximately 13.35% of miniature poodles have a height exceeding 14 inches.

Step-by-step solution

Calculate the z-score

First, we need to calculate the z-score for the given height, 14 inches. The z-score formula is: \(z = \frac{X - \mu}{\sigma}\) where: - \(z\) = z-score - \(X\) = given height (14 inches) - \(\mu\) = mean height (12 inches) - \(\sigma\) = standard deviation (1.8 inches) Plugging the values into the formula: \(z = \frac{14 - 12}{1.8}\)

Evaluate the z-score

Now, we need to evaluate the z-score we have just calculated: \(z = \frac{14 - 12}{1.8} = \frac{2}{1.8} = 1.11\) So the z-score for a height of 14 inches is approximately 1.11.

Find the probability of the z-score

We need to find the percentage of poodles with a z-score greater than 1.11. Using a standard normal distribution table or calculator, we can find the probability associated with the calculated z-score: Probability for z = 1.11: P(z > 1.11) = 1 - P(z ≤ 1.11) = 1 - 0.8665≈ 0.1335

Convert the probability to percentage

Finally, we will convert the probability (0.1335) to a percentage by multiplying by 100: Percentage = 0.1335 * 100 ≈ 13.35% Thus, approximately 13.35% of miniature poodles have a height exceeding 14 inches.