Question 1:
New spark plugs have just been installed in a small airplane with a four-cylinder engine. There is one spark plug per cylinder, so four spark plugs have just been installed. For each spark plug, the probability that it is defective and will fail during its first 20 minutes of flight is 1/10,000, independent of the other plugs.
(a) For each spark plug, what is the probability that it will not fail?
(b) If only one spark plug fails, the plane will shake and not climb higher, but it can be landed safely. What is the probability that this happens? Calculate using the appropriate formula.
(c) If two or more spark plugs fail, the plane will crash. What is the probability that this happens? You may use software to help determine this probability.
(d) Given that one spark plug fails, what is the probability that the plane will still land safely?
Question 2:
A particular brand of sugar is sold in bags labelled 1kg. The actual weights are approximately normally distributed with a mean of 1004g and a standard deviation of 2.9g.
(a) What is the probability of a customer being sold an underweight (less than 1kg) bag of sugar?
(b)
(i) Find the Z score corresponding to the bottom 4% of the N(0,1) distribution.
(ii) Assuming the filling variability achievable by the packaging machine cannot be improved, what would the mean bag weight need to be to ensure that no more than 4% of customers are sold an underweight bag?
Question 3:
When certain quantities are measured, the last digits tend to be uniformly distributed, but if they are estimated or reported, the last digits tend to have disproportionately more 0s and 5s. The table below is based on counts of the last digits in a set of weights reported for a sample of 80 people. Conduct an appropriate hypothesis test to assess the claim that the digits come from a population where they each have the same probability of occurring. Does it appear that these weights were measured or estimated?
Last Digit 0 1 2 3 4 5 6 7 8 9
Frequency 10 7 4 11 6 10 5 9 10 8
(a) State the relevant null and alternative hypotheses.
(b) Calculate the appropriate test statistic. (By hand or using JMP.)
(c) Determine the p-value. (Using tables, probability calculator software or from JMP output.)
(d) Give a decision about the null hypothesis.
(e) Make an appropriate conclusion.