The length of rose stems follows a normal distribution with a mean length of 17.55 inches and a standard deviation of 3.627 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 19.7 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 21.7 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 4% 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 0.93. Which of the following statements is a correct interpretation of the meaning of this value?
i. The length of this stem is 0.93 standard deviations longer than the average rose stem.
ii. The length of this stem is 0.93 times longer than the average rose stem.
iii. The length of this stem is 0.93 inches longer than the average rose stem.
iv. There is not enough information provided to interpret this value.

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

The first advantage of the normal distribution is that it is symmetric and bell-shaped. This shape is useful because it can be used to describe many populations, from classroom grades to heights and weights.

Normal distributions are symmetric, unimodal, and asymptotic, and the mean, median, and mode are all equal. A normal distribution is perfectly symmetrical around its center.

The probability calculations are as follows:

a. The probability that a rose stem will be shorter than 19.7 inches is 0.7978.

b. The probability that a rose is considered a long stem rose if its length is over 21.7 inches is 0.0717.

c. The maximum stem length for a rose to qualify as a corsage is 12.7493 inches.

d. The correct interpretation of a Z-score of 0.93 is that the stem length is 0.93 standard deviations longer than the average rose stem length.