| MTH219: Fundamentals of Statistics and Probability |
Question 1
A test, in which the marks obtainable range from 0 to 100 inclusive, was taken by 30 students who had been trained in such tests, and by 20 other students who had not. A summary of the results is given in Table Q1 below.
(a) Calculate the mean and standard deviation of the combined set of 50 scores. Give your answers to two decimal places.
(b) One of the students was ill but took the test and scored only 1 mark. It was decided to exclude this result from the analysis.
(i) Calculate the mean and standard deviation of the remaining 49 scores.
Give your answers to two decimal places.
(ii) Briefly explain what would be the changes in the mode and median when
compared to those of the full set of 50 scores.
Question 2
In a batch of manufactured items, the proportion of defective items is p. From each batch a random sample of nine is taken and tested. If two or more items are found to be defective, the batch is rejected, otherwise it is accepted.
(a) Show that the probability that a batch is accepted is (1 − 𝑝) 8 (1 + 8𝑝).
(b) It is decided to modify the checking scheme such that when one defective is found in the sample, a second sample of nine is taken and the batch rejected if this contains any defectives. With this exception, the original scheme is continued.
(i) Apply the modified scheme and derive an expression in terms of p for the
the probability that a batch is accepted, simplifying your answer as much as
possible.
(ii) Evaluate and comment on the average number of items sampled over a
large number of batches, with this new sampling scheme, when p = 0.1.
Provide your answer to 2 decimal places.
Question 3
A drink dispensing machine dispenses either cups of tea or coffee. The number of cups of tea sold may be assumed to be a Poisson variable with mean of 0.5 cups per 10 minutes period, while the number of cups of coffee sold may be assumed to be a Poisson variable with mean of 1.5 cups per 10 minutes period.
(a) Compute the probability that in a given half-hour period, exactly 2 cups of tea and 2 cups of coffee are dispensed by the dispensing machine.
(b) Calculate the probability that in a given 45 minutes period, more than 7 drinks are dispensed by the dispensing machine.
(c) In a 10 minutes period, 3 drinks were dispensed by the dispensing machine.
Calculate the probability that these were all cups of coffee.
Question 4
During peak hours, the number of bus “74” arriving at the bus stop outside SUSS campus is a Poisson variable with rate of 9 buses per hour.
(a) Calculate the probability that you would need to wait more than 20 minutes for the bus “74” if one bus “74” just departed on your arrival at the bus stop. Show full details of your workings.
(b) Briefly explain if you would be expected to wait longer at the bus stop if the bus “74” has just departed when you arrive at the bus stop.
Question 5
In a city, the mean height of married man is 180 cm with a standard deviation of 4 cm, the mean height of married women is 175 cm with a standard deviation of 3 cm. Assume that the choice of partner in marriage is not influenced by height consideration. A couple is selected at random. Showing all details of your workings for the following computations.
(a) Compute the probability that
(i) both of them are taller than 177.5cm;
(ii) the husband is taller by less than 5cm;
(iii) their height difference is less than 5cm.
(b) 7 more married couples are selected from the city. Compute the probability that at least 2 of the selected couples have height differences of more than 5cm.
Question 6
The life in days, X, of an insect is such that log10 𝑋 is normally distributed with a mean of 2 days and a standard deviation of 0.2 days.
(a) Calculate the probability that an insect will have a life of
(i) more than 200 days;
(ii) between 50 and 150 days inclusive.
(b) Two insects have life expectancies of T1 and T2, and 𝐻
(i) Show and describe the distribution type of log10 𝐻, if their life
expectancies are independent.
(ii) If one insect has a life expectancy 1.8 times that of the other, is this
Is the difference significant?
