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. 

(a) [2 marks] 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)? 

(b) [2 marks] How much must a Roma tomato weigh to be among the heaviest 20%? 

(c) [2 marks] 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? 

(d) [3 marks] 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. 

(e) [3 marks] 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?