1 A student must pass through two sets of traffic lights on his way to university each day. 1 The first light is green 35% of the time, yellow 8% of the time and red 57% of the time. The second light is green 44% of the time, yellow 11% of the time and red 45% of the time. Suppose that it is known that the two sets of lights operate independently. (a) On any given trip to university, what is the probability that both lights will be the same colour when the student arrives at the intersections? (b) What is the probability that the first light will be green or that the second light will be red? (c) Let X be the number of times the student stops at a light on his way to school. (He will only stop if the light is red, and not if it is yellow.) Find the probability distribution of X. Question 2 According to the 2011 Canadian census, the median income of households in Canada is 2 $72,240. In a random sample of 500 Canadian households, the median income was calculated to be $68,179. It is known that 19% of households in Canada are found in rural areas, whereas 110 of the households in the sample (22%) were from rural areas. Determine whether the values in bold are parameters or statistics? Question 3 Determine whether the variable X has a binomial distribution in each of the following 3 cases. If it does, explain why and determine the values of the parameters n and p. If it doesn't, explain why not. (a) You randomly select one card from each of ten different decks of playing cards. X = number of Clubs selected (b) A boy asks the same girl to go on a date every day for a week. X = number of times the girl agrees to go on a date with the boy (c) You toss four fair coins -- a loonie, a quarter, a dime and a nickel. X = number of coins that land on Heads (d) A fair die is repeatedly rolled. X = number of rolls required to observe the number 5 for the third time (e) A fair die is rolled five times. X = number of times the number 3 is observed. (f) You are dealt a five-card poker hand. X = number of Diamonds in your hand (g) Weights of oranges sold at a supermarket follow a normal distribution with mean 175 grams and standard deviation 20 grams. You take a random sample of 7 oranges. X = number of oranges that weigh more than 200 grams (h) An airplane carrying 100 passengers has two emergency exits -- one at the front of the plane and one over the wings. The plane makes an emergency landing. X = number of passengers that exit the plane through the front emergency exit Question 4 A student must pass through 12 sets of traffic lights on his way to university. Suppose that each of the lights is green 35.7% of the time, yellow 5.4% of the time, and red 58.9% of the time. Suppose also that the traffic lights function independently. (a) Let X be the number of red lights encountered by the student on his way to university. Find the mean and variance of X. (b) What is the probability that the student encounters exactly five red lights on his way to university one day? (c) What is the probability that the student encounters at least three green lights on his way to university one day? (d) In a five-day school week, what is the probability that the student encounters exactly six yellow lights on his way to university? Question 5 5 Daily water intake (including water used in drinks such as coffee, tea and juice) for Canadian adults follows a normal distribution with mean 1.82 litres and standard deviation 0.3 litres. (a) Can you calculate the probability that the mean daily water intake for a random sample of two Canadian adults is less than 1.61 litres? If you can, then calculate it. If you can't, explain why not. (b) What is the probability that the mean daily water intake for a random sample of 33 Canadian adults is greater than 1.76 litres? (c) What is the probability that the total daily water intake for a random sample of 46 Canadian adults is between 80.36 and 85.33 litres? (d) Using the 68-95-99.7 Rule, approximately 95% of samples of 23 Canadian adults will have mean daily water intakes between ________ and ________ . (e) If we take a random sample of 11 Canadian adults, there is an 11% chance that their mean daily water intake will be greater than ________. (f) Are the probabilities you calculated in (b) and (c) exact or approximate? Explain. Question 6 service times for customers at a post office follow some right-skewed distribution with mean 2.97 minutes and standard deviation 1.82 minutes. (a) Can you calculate the probability that the average service time for the next two customers is less than 2.7 minutes? If you can, then calculate it. If you can't, explain why not. (b) What is the probability that the mean service time for a random sample of 31 customers is between 2.5 and 3.19 minutes? (c) What is the probability that the total service time for the next 48 customers exceeds 161.73 minutes? (Assume that the next 48 customers represent a simple random sample and that there is always a customer waiting in line.) (d) What is the probability that the mean service time for a random sample of 58 customers is exactly 3.65 minutes? (e) Are the probabilities you calculated in (b) and (c) exact or approximate? Explain. Question 6 options