Question

Agricultural scientists are working on developing an improved variety of a breed of tomatoes. Marketing research indicates that customers are likely to bypass tomatoes that weigh less than 72 grams. The current variety of tomatoes produces fruit that averages 76 grams, but 10​% of the tomatoes are too small. It is reasonable to assume that a normal model applies. Complete parts a through d below.

what is the standard deviation of the weights of tomatoes being grown?

Answer 1

To find the standard deviation of the weights of the tomatoes, we use the normal distribution properties. We calculate the Z-score for the 10th percentile and use the values (72 grams for X, 76 grams for μ, and -1.28 for Z) to find a standard deviation of approximately 3.125 grams.

Step-by-step explanation:

The question is asking us to find the standard deviation of the weights of tomatoes being grown, based on the given information that 10% of the tomatoes weigh less than 72 grams and the average weight of the tomatoes is 76 grams, under the normal distribution model.

To solve this, we can use the normal distribution properties, where the cutoff point representing the smallest 10% of weights under the normal curve corresponds to a Z-score of -1.28 (using standard Z-score tables or a calculator). In this context, the Z-score is calculated using the formula (X - μ) / σ, where X is the value of interest (72 grams), μ is the mean (76 grams), and σ is the standard deviation.

Re-arranging the Z-score formula to solve for σ, we get σ = (μ - X) / Z. Substituting the known values gives us σ = (76 - 72) / -1.28, which leads to a standard deviation of approximately 3.125 grams. Hence, the standard deviation of the weights of tomatoes being grown is 3.125 grams.