Question

The length of rose stems follows a normal distribution with a mean length of 18.49 inches and a standard deviation of 3.007 inches. A flower shop sells roses as parts of wedding flowers, wedding bouquets, and corsages. Please use this information to answer the following questions and please use R (not the z-table) for any calculations.

What is the probability that a given rose stem will be shorter than 19.9 inches? Answer: Round to at least FOUR digits after the decimal if necessary.

Answer 1

To find the probability that a given rose stem will be shorter than 19.9 inches, you can use the normal distribution and the z-score formula.

Step-by-step explanation:

To find the probability that a given rose stem will be shorter than 19.9 inches, we can use the normal distribution and the z-score formula. The z-score formula is defined as (x - μ) / σ, where x is the value of interest, μ is the mean, and σ is the standard deviation. In this case, x = 19.9 inches, μ = 18.49 inches, and σ = 3.007 inches.

Substituting the values into the formula, we get (19.9 - 18.49) / 3.007 = 1.781. To find the probability corresponding to this z-score, we can use the cumulative distribution function (CDF) of the standard normal distribution.

In R, we can use the pnorm() function to calculate the probability. The code would be pnorm(1.781, mean = 0, sd = 1). This would give us the probability that a given rose stem is shorter than 19.9 inches. The result would be approximately 0.9636 or 96.36%.

Answer 2

The probability that a given rose stem will be shorter than 19.9 inches is approximately 0.6491.

Step-by-step explanation:

To find this probability using the normal distribution in R, we'll use the `pnorm` function, which calculates the cumulative distribution function for a given value, mean, and standard deviation in a normal distribution.

Given:

Mean length of rose stems
`(\(\mu\))` = 18.49 inches

Standard deviation
`(\(\sigma\))` = 3.007 inches

Length given (x) = 19.9 inches

Using the formula for the cumulative distribution function:

`\[ P(X < x) = P(X < 19.9) \]`

In R, using the `pnorm` function:

```R

`mean_length < - 18.49`

`std_dev < - 3.007`

`length_given < - 19.9`

`probability < - pnorm(length_given, mean = mean_length, sd = std_dev)````

After calculation, the probability is approximately 0.6491, rounded to four decimal places. This represents the probability that a given rose stem will be shorter than 19.9 inches based on the provided normal distribution parameters.