Question 1: The General Cereal Company believes that at least 20% of customers prefer its cereal to other brands of cereal. In a sample of 300 cereal eaters, 55 preferred the General Cereal Company brand to other brands.
a) Construct a 96% confidence interval for the true proportion of cereal eaters who prefer General brand cereal.
b) If the General Cereal Company wants to estimate the proportion of cereal eaters who would prefer a new, and yet untested all-grain cereal, how large a sample is required to determine the proportion who prefer this all-grain cereal to within 0.02 and with a 97% level of confidence?
Question 2: A tire manufacturer is considering a newly designed tread pattern for its all-weather tires. The Design and Development Division claims that the tire will allow a car travelling at 110 km/h to come to a complete stop in less than an average of 40 meters after brakes are applied. The Product Evaluation Department tests 10 sets of the new tires on vehicles on the test track and records the following stopping distances (in meters) for vehicles travelling at 110 km/h.
42.1 39.9 39.6 40.2 39.2 37.4 36.1 37.8 39.0 39.6
a) Test the claim of the Design and Development Division at a significance level of 0.05. Clearly show the steps of the hypothesis test that you use, and express your final conclusion using a clear English sentence. Assume that for a given type of tire the stopping distance of cars is normally distributed.
b) If the tire manufacturer wants to reassess the stopping distance of a newly designed tire, how large a sample is required to assess the stopping distance to within 0.50 meters, and with a significance level of 0.05. Assume the standard deviation of the population is 2.8 meters.
Question 3: In the 1980's, it was generally believed that congenital abnormalities affected about 5% of the nation's children. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of abnormalities. A recent study examined 425 children and found 42 of them showed signs of abnormalities. Is this evidence that the risk has increased? Test your claim at a significance level of 0.01. Clearly show the steps of the hypothesis test that you use, and express your final conclusion using a clear English sentence.
Question 4: A Canadian railway company claims that its trains block a railway crossing in a rural community for no more that 5 minutes per train on the average. A local citizens group doubts this claim and measure the crossing times of a random selection of trains. The actual times (in minutes) that 16 randomly selected trains block the crossing were:
5.70 5.35 4.55 5.50 5.15 6.35 4.60 6.00
5.10 5.00 4.25 4.80 5.20 5.60 5.90 5.40
a) Assuming the times the trains block the crossing are normally distributed, what is the margin of error for the estimate of the time the trains block the railway crossing, assuming a significance level of 0.05?
b) Why is it necessary to be told in this question that the distribution of times is normally distributed?
c) What is the 96% confidence interval for the estimate of the time the trains block the railway crossing?
d) What is the value of the test statistic when testing the claim of the local citizens group?
e) What is the critical value one uses when testing the claim of the local citizens group?