Question

Market research has indicated that customers are likely to bypass Roma tomatoes that weigh less than

70 grams. A produce company produces Roma tomatoes that average 78.0 grams with a standard

deviation of 5.2 grams.

i) Assuming that the normal distribution is a reasonable model for the weights of these

tomatoes, what proportion of Roma tomatoes are currently undersize (less than 70g)?

ii) How much must a Roma tomato weigh to be among the heaviest 20%?

iii) The aim of the current research is to reduce the proportion of undersized tomatoes

to no more than 2%. One way of reducing this proportion is to reduce the standard deviation.

If the average size of the tomatoes remains 78.0 grams, what must the target standard deviation

be to achieve the 2% goal?

iv) The company claims that the goal of 2% undersized tomatoes is reached. To test this,

a random sample of 20 tomatoes is taken. What is the distribution of the number of undersized

tomatoes in this sample if the company's claim is true? Explain your reasoning.

v) Suppose there were 3 undersized tomatoes in the random sample of 20. What is the

probability of getting at least 3 undersized tomatoes in a random sample of 20 if the company's

claim is true? Do you believe the company's claim? Why or why not?

Answer 1

Let X is a random variable shows the weight of tomatoes. Here X has normal distribution with following parameters

i)

The z-score for X = 70 is

Using z table, the proportion of Roma tomatoes are currently under size is

Answer: 0.0618 or 6.18%

ii)

Here we need to find X such that;

P(X >= x) = 0.20

For that we need z-score that has 0.20 area to its right. Using z table, the z-score 0.84 has 0.20 area to it right and 1 - 0.20 = 0.80 area to its left.

The required weight using z -score formula is

Answer: 82.37

(iii)

Standard deviation needed such that

For that we need z-score that has 0.02 area to its left. Using z table, the z-score -2.05 has 0.02 area to its left. So required population standard deviation is

Answer: 3.90

(vi)

Sample size: n= 20

Let Y is a random variable shows the number of undersized tomatoes. Here Y has binomial distribution with parameters as follows:

n=20, p = 0.02

v)

The probability of getting at least 3 undersized tomatoes in a random sample of 20 if the company's claim is true is

_{Answer: 0.0071}