Question 1. When a truckload of apples arrives at a packing plant, a random sample of 150 is selected and examined for bruises, discoloration, and other defects. The whole truckload will be rejected if more than 5% of the sample is unsatisfactory. Suppose that 8% of the apples on the truck do not meet the desired standard. What is the probability that the shipment will be accepted anyway ?
Question 2. The average January temperature in Calgary varies from year to year. These average January temperatures themselves have an average, over many past years, of -7.9 degrees Celsius, with a standard deviation of 5.0 degrees. Assume a Normal model applies.
a) During what percentage of years does Calgary have a January average above 0 degrees ?
b) In the 20% coldest Januaries, the average January temperature reaches no higher than how many degrees ?
c) Suppose you will be in Calgary for the next four years. Describe the sampling distribution model for the average of the next four January average temperatures. Assume climatic conditions are not changing.
d) What is the probability that the average of the next four January average temperatures will exceed -4 degrees ? If this happens, would you consider this evidence of climactic warming (at least locally in Calgary), or just Normal variations ?
Question 3. STAT2303 is a large introductory statistics course at the University of Toronto, with 720 students enrolled. On the term test, the average score was 72, with a standard deviation of 16. There are 24 tutorials, with 30 students in each. David teaches one and Jonathan teaches another. Assume that tutorial groups are formed quite randomly.
a) We select a student at random from Jonathan's tutorial. Find the probability that this student's test score is at least 80. Assume that scores are Normally distributed.
b) We select one student at random from each tutorial. Find the probability that Jonathan's student outper- formed David's student by at least 7 marks on the test. Assume that scores are Normally distributed. (Hint: work with the di erence between two scores)
c) What is the probability that the average for Jonathan's tutorial is at least 80 ? Do you need to assume that scores are Normally distributed?
d) What is the probability that the average test score in Jonathan's tutorial is at least seven marks higher than the average test score for David's tutorial? Do you need to assume that scores are Normally distributed ?