MTH219: A test, in Which the marks Obtainable Range from 0 to 100 inclusive, was Taken by 30 Students: Fundamentals of Statistics and Probability Assignment, SUSS, Singapore

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, the second sample of nine is taken and the batch is 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 a 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 a 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 the SUSS campus is a Poisson variable with the 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 a 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?

1) Suppose X is a discrete random variable with the following information:

The range of all possible values of X are: 0, 1, 2, 3 and 4

P(X = 3) = 0.33      P(X = 1) = 0.11

P(X = 0) = 0.08     P(X = 2) = 0.27

Determine the value of  F(2) and F(4) respectively.

2) Suppose X is a discrete random variable with the following information:

The range of all possible values of X are: 0, 1, 2, 3 and 4

P(X = 3) = 0.33      P(X = 1) = 0.11

P(X = 0) = 0.08     P(X = 2) = 0.27

Find the expected value of X.

3) Suppose X is a discrete random variable with the following information:

The range of all possible values of X are: 0, 1, 2, 3 and 4

P(X = 3) = 0.33      P(X = 1) = 0.11

P(X = 0) = 0.08     P(X = 2) = 0.27

Find the variance of X.

4) In a Prime Minister’s press conference, 50 invited reporters are each assigned a seat in the conference room with 50 pre-arranged socially distanced seats. The reporters then take their seats in a predetermined order.

The first reporter to take a seat has forgotten his/her assigned seat and takes a seat at random. Each subsequent reporter will then take his/her assigned seat unless it is already occupied, in which case, an unoccupied seat is chosen at random.

What is the probability that the last of the 50 reporters will be seated at his/her assigned seat? Give your answer to 2 decimal places

5) A large table is covered with a table cloth with a black-white checker-box pattern (with alternating black and white squares like those on a chessboard), each square is 5cm by 5cm.

If a coin of diameter 2cm is tossed onto the table, what is the probability that the coin touches one or more of the black squares?

Assume the coin lands on the large table. Give your answer to 2 decimal places.

6) X is a random variable such that

P[X = 1] =r, P[X = 0] = 1 – r,      where r> 0.5

X₁, X₂, X₃ and X₄ are independently and identically distributed random variables, each with the same distribution as X. We define Ywhere

Y = X₁ + X₂ + X₃ + X₄

If var(X) = 3/16 (= 0.1875), calculate P[Y ≥ 2]. Give your answer to 2 decimal places.

24) Given that the probability that an archer hits the target with each arrow he shoots is 0.85. The archer shoots arrows repeatedly until he hits the first target. What is the expected value number of arrows he shoots?