Question

A strain of long stemmed roses has an approximate normal distribution with a mean stem length of 15 inches and standard deviation of 2.5 inches. a. If one accepts as "long-stemmed roses" only those roses with a stem length greater than 12.5 inches, what percentage of such roses would be unacceptable? b. What percentage of these roses would have a stem length between 12.5 and 20 inches?

Short answer

Answer: Approximately 15.87% of roses have a stem length less than 12.5 inches, and approximately 81.85% of roses have a stem length between 12.5 and 20 inches.

Step-by-step solution

Find the Z-score formula

To find the Z-score for a given stem length in a normal distribution, we need to use the following formula: Z = (X - μ) / σ Where: Z is the Z-score X is the stem length μ is the mean (15 inches) σ is the standard deviation (2.5 inches)

Calculate the Z-scores

Now that we have the formula, let's calculate the Z-score for the given stem lengths. For part a: X = 12.5 inches Z = (12.5 - 15) / 2.5 = -1 For part b: X1 = 12.5 inches, X2 = 20 inches Z1 = (12.5 - 15) / 2.5 = -1 Z2 = (20 - 15) / 2.5 = 2

Use the Z-table to find the percentages

Now that we have the Z-scores, we can find the percentage of roses by using the Z-table. For part a: The Z-table value for Z = -1 is 0.1587. This means that approximately 15.87% of roses have a stem length of less than 12.5 inches and would be considered unacceptable. For part b: The Z-table value for Z1 = -1 is 0.1587. The Z-table value for Z2 = 2 is 0.9772. To find the percentage of roses with stem lengths between 12.5 and 20 inches, we subtract the Z-table value for Z1 (-1) from the Z-table value for Z2 (2): 0.9772 - 0.1587 = 0.8185. Thus, approximately 81.85% of roses have stem lengths between 12.5 and 20 inches. So, for the exercise: a. Approximately 15.87% of roses would be unacceptable. b. Approximately 81.85% of these roses would have a stem length between 12.5 and 20 inches.

Key concepts

Z-score

A Z-score is a way of describing the position of a particular value within a normal distribution. In simpler terms, it tells us how many standard deviations a given value is from the mean of the distribution.
To calculate the Z-score for a particular observation, use the formula: \[ Z = \frac{X - \mu}{\sigma} \] Here, \(X\) is the value we’re interested in, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation.
This formula converts a normal distribution value into a standard normal distribution, making it easier to determine the probability or the percentage of data points below or above \(X\).
When the Z-score is negative, it indicates that the value is below the mean. Conversely, a positive Z-score indicates a value above the mean.

Standard Deviation

The standard deviation is a statistical measure that describes the amount of variation or dispersion in a set of values. In the context of normal distribution, it tells us how spread out the observations are from the mean.
A smaller standard deviation means that the values are closer to the mean, while a larger standard deviation means there is more variation among the values.
To calculate the standard deviation, you typically start by finding the variance (the average of squared differences from the mean) and then take the square root of the variance.
In our roses problem, the standard deviation is 2.5 inches, which helps us understand the variability in stem length. It is a crucial component in calculating the Z-scores.

Percentage Calculation

Percentage calculation in this context involves determining what proportion of the data set falls into specific ranges, using Z-scores and a Z-table.
In our problem, we calculated the Z-scores for different stem lengths and used these scores to find percentages of roses fitting certain criteria.
For instance, if a particular Z-score is -1, using the Z-table reveals that approximately 15.87% of the data falls below this value. This means that such a percentage of roses would have an unacceptable stem length.
When we wanted to know what percentage of rose stems were between two lengths, we subtracted one percentage from the other. This simple subtraction gives us the proportion of data points within a specified range.

Mean

The mean, often referred to as the average, is a measure of central tendency. It is calculated by summing all values in a data set and dividing by the number of observations.
In the context of our example, the mean stem length of the roses is 15 inches. This number represents the central point of the distribution, around which the stem lengths are distributed.
The mean is a critical parameter when calculating Z-scores, as it allows us to determine how far each observation is from the central tendency of the data.
Understanding the mean also helps in visualizing where the bulk of data is likely to sit within a normal distribution.

Z-table

The Z-table, or standard normal table, is a tool used to interpret Z-scores by providing the probability that a standard normal random variable is less than or equal to a given Z-score.
It lists the percentage of the area under the normal curve to the left of a specific Z-score. This percentage is synonymous with the probability of data falling below the given Z-score.
In our example, after calculating Z-scores for the stem lengths, we used the Z-table to determine how much of the data lies below certain lengths (e.g., below 12.5 inches).
This table is instrumental in converting a Z-score to a meaningful probability, enabling the calculation of percentages like in our roses' problem.