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The length of rose stems follows a normal distribution with a mean length of 19.86 inches and a standard deviation of 3.853 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 use R (not the z-table) for any calculations.

a. What is the probability that a given rose stem will be shorter than 21.9 inches?
Answer: Round to at least FOUR digits after the decimal if necessary.

b. Suppose a rose is considered a 'long stem rose' if its stem length is longer than 24 inches. What is the probability that a given rose will be considered a long stem rose?
Answer: Round to at least FOUR digits after the decimal if necessary.

c. The flower shop has a rule that the shortest 6% of roses are clipped and used as corsages. What is the maximum stem length (in inches) a rose can be and still qualify to be used as a corsage by the shop?
Answer: _______ inches. Round to at least FOUR digits after the decimal if necessary.

d. Suppose the z-score (standardized score) of a rose stem length is given as 0.89. Which of the following statements is a correct interpretation of the meaning of this value?
A. The length of this stem is 0.89 times longer than the average rose stem.
B. The length of this stem is 0.89 inches longer than the average rose stem.
C. The length of this stem is 0.89 standard deviations longer than the average rose stem.
D. There is not enough information provided to interpret this value.

Answer :

Final answer:

To answer these questions, we use the normal distribution and R programming. We calculate z-scores to find probabilities and to determine cutoff points. We can use the 'qnorm' function in R to find values corresponding to specific percentiles. The z-score of 0.89 means the length of the stem is 0.09 standard deviations longer than the average rose stem.

Explanation:

To answer these questions, we can use the normal distribution. Let's break down each question:

a. To find the probability that a given rose stem will be shorter than 21.9 inches, we need to calculate the z-score and find the corresponding probability using R. The formula to calculate the z-score is:

z = (x - mean) / standard deviation

where x is the given value, mean is the mean length of rose stems, and standard deviation is the standard deviation of rose stem lengths. Using R, we can calculate the z-score and find the probability that a rose stem will be shorter than 21.9 inches.

b. To find the probability that a given rose will be considered a long stem rose (stem length > 24 inches), we follow a similar process as in part a. Calculate the z-score and find the probability.

c. To find the maximum stem length that qualifies to be used as a corsage, we need to find the value of x (stem length) that corresponds to the 6th percentile. We can use R to find this value by first calculating the z-score using the formula mentioned earlier, and then using the 'qnorm' function in R to find the corresponding value of x.

d. The z-score of 0.89 tells us that the length of this stem is 0.09 standard deviations longer than the average rose stem. So, the correct interpretation is option C.

Learn more about Normal Distribution here:

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