BUSINESS DATA ANALYSIS, A17, TEST 3

Transcription

1 BUSINESS DATA ANALYSIS, A17, TEST 3 Name: Student number (1) (2.5 marks) In healthy adults, the ph of arterial plasma (a measure of acidity of the blood) is µ = 7.4. A new drug for arthritis has been developed. However, it is thought that this drug may change blood ph. A random sample of 31 patients with arthritis took the drug for 3 months. Blood tests showed that in the sample, the average ph was 8.1 with a sample standard deviation of 1.9. At the 5% level of significance, does the sample data indicate that the patients taking the drug had a mean ph larger than 7.4? Test H 0 : µ = 7.4 versus H 1 : µ > 7.4. What is the P -value? Write a conclusion in an English sentence.

2 2 BUSINESS DATA ANALYSIS, A17, TEST 3 (2) (2.5 marks) In the Volkswagen emissions scandal (also called dieselgate ), it was revealed that the car manufacturer Volkswagen had equipped many of its models with test-rigging software that would make their emissions appear much lower than what they were under real-world driving conditions. The scandal led to millions of recalls, a sharp decline in sales, and a dramatic drop in Volkswagen share prices. An early independent study of the real nitrogen oxide emissions of the VW Passat showed that in a sample of 10 cars, the average emission was 0.58 with a sample standard deviation of 0.18 (both in grams per kilometer). (a) Construct a 98% confidence interval for the average emission of all VW Passat cars. (b) Can the manufacturer reasonably claim that the VW Passat satisfies the Euro 5 emission standard, which prescribes a maximum nitrogen oxide emission of 0.29 grams per kilometer? Justify briefly.

3 BUSINESS DATA ANALYSIS, A17, TEST 3 3 (3) (3 marks) The National Garlic Farmer s Association would like to estimate how much they could earn for their garlic crop in the month of November. For random sample of 40 farming regions, the sample mean was dollars per 200 pounds of garlic. Assume that the population standard deviation is known to be σ = dollars per 200 pounds of garlic. (a) Construct an 90% confidence interval for the population mean price (per 200 pounds) that farmers would receive for their garlic crop. (b) Find the sample size necessary to construct a 95% confidence interval with a margin of error of 40$.

4 4 BUSINESS DATA ANALYSIS, A17, TEST 3 (4) (3 marks) As the world s largest passenger aircraft, the Airbus A380 is certified to carry up to 868 passengers (538 on the main deck and 330 on the upper). The budget airline CheapoAir is operating such 868-seat aircraft on their busy holiday route from London Heathrow to Palma de Mallorca. In order to maximize profit, CheapoAir sell 950 tickets per flight, knowing from experience that not all ticketed passengers show up for their flight. Assume that the probability that a passenger does not show up is 8%, and passengers behave independently. (a) Calculate the expected value and the standard deviation for the number of people who show up for the flight. (b) Estimate the probability that between 840 and 850 people (inclusive) will show up. (c) Estimate the probability that every passenger who shows up can take the flight (that is, at most 868 people show up).

5 BUSINESS DATA ANALYSIS, A17, TEST 3 5 (5) (3 marks) A manufacturer of flick knives claims that their blades can go 60 days without sharpening. Quality control took 7 knives off the production line and tested them for several months. Below are the number of days that the knives went for before losing their edge (a) At the 2% level of significance, test whether the average number of days before the knives require sharpening is less than 60 days. Make sure to include an estimate for the p-value for the test. (b) Write an appropriate conclusion in the context of the problem.

6 6 BUSINESS DATA ANALYSIS, A17, TEST 3 (6) (3 marks) The birth weight of newborn babies in Canada is approximately normally distributed, with a mean of 3.35kg and standard deviation 0.55kg. (a) What is the probability that a newborn baby weighs between 3kg and 4kg? (b) How much do the babies in the heaviest quartile weigh? (c) Find the weight range of the middle 80% of newborn babies. (d) What is the probability that the first 100 babies born in 2018 will have a (sample) average weight of more than 3.5kg?

7 BUSINESS DATA ANALYSIS, A17, TEST 3 7 (7) (3 marks) In a poll conducted in September 2016 by the Angus Reid Forum in partnership with the national broadcaster CBC, a representative randomized group of Canadian adults were asked about their feelings about the independence of Quebec. Of the 3904 people surveyed, 3194 agreed with the statement that Ultimately, Quebec should stay in Canada. (a) Construct a 95% confidence interval for the true proportion of Canadians who feel that Ultimately, Quebec should stay in Canada. (b) What minimum sample size would be required to estimate the proportion of Canadians who feel that Ultimately, Quebec should stay in Canada if we wanted to be 98% confident that the distance between the estimate ˆp and the true proportion p is no more than 1 percentage point? (c) Suppose you want to design a survey to find out how many Canadians believe that Alberta should not stay in Canada, with 95% confidence, and a margin of error of no more than 2 percentage points. What minimum sample size would you use? (You do not have a preliminary estimate of the proportion in this case.)

8 8 BUSINESS DATA ANALYSIS, A17, TEST 3 Formulas z = X µ, X = µ + zσ σ µ = np, σ = np(1 p), z = (X ± 0.5) µ σ x z α/2 σ n < µ < x + z α/2 σ n n = z2 α/2 σ2 E 2 s s x t α/2 < µ < x + t α/2 n n ˆpˆq ˆpˆq ˆp z α/2 n < p < ˆp + z α/2 n, ˆq = 1 ˆp n = 0.25z2 α/2 E 2, n = ˆpˆqz2 α/2 E 2, z = x µ σ/ n, t = x µ s/ n