Question

jardins garden
mr. jardin wants to know how many tomato plants he needs to start with in february to have 100 plants to sell this spring at the local market.

• by spring - time, the height of tomato plants in mr. jardins garden are normally distributed with a mean of 48.8 inches and a standard deviation of 3 inches.
• mr. jardin only sells plants that are between 42.8 and 54.8 inches tall.

1. use the empirical rule to estimate the percentage of mr. jardins tomato plants that will be between 42.8 and 54.8 inches tall.
2. how many tomato plants would you tell mr. jardin he should start with? explain.

Explanation:

Step1: Recall the Empirical Rule

The Empirical Rule for a normal - distribution states that approximately 68% of the data lies within 1 standard deviation of the mean, 95% lies within 2 standard deviations of the mean, and 99.7% lies within 3 standard deviations of the mean. First, find the number of standard deviations from the mean for the given values.
Let $\mu = 48.8$ inches and $\sigma=3$ inches. For $x_1 = 42.8$ inches, $z_1=\frac{x_1 - \mu}{\sigma}=\frac{42.8 - 48.8}{3}=\frac{- 6}{3}=-2$. For $x_2 = 54.8$ inches, $z_2=\frac{x_2 - \mu}{\sigma}=\frac{54.8 - 48.8}{3}=\frac{6}{3}=2$.

Step2: Estimate the percentage

Since the values 42.8 and 54.8 are 2 standard deviations below and above the mean respectively, by the Empirical Rule, approximately 95% of Mr. Jardin's tomato plants will be between 42.8 and 54.8 inches tall.

Step3: Calculate the number of plants to start with

Let $N$ be the number of plants to start with. We know that the proportion of plants that are sell - able is 0.95, and we want to have 100 sell - able plants. We set up the proportion $\frac{100}{N}=0.95$. Solving for $N$, we get $N=\frac{100}{0.95}\approx105.26$. Since we can't have a fraction of a plant, Mr. Jardin should start with 106 plants.

Answer:

1. Approximately 95%
2. Mr. Jardin should start with 106 plants. We know that 95% of the plants will be within the sell - able height range. Let the number of plants to start with be $N$. Using the proportion $\frac{100}{N}=0.95$, solving for $N$ gives $N=\frac{100}{0.95}\approx105.26$. Since we need whole plants, we round up to 106.