MTH219e- Suppose an Undergraduate-Admissions Committee Rated 400 Applicants : Fundamentals of Statistics and Probability Tutor-Marked Assignment, SUSS, Singapore

MTH219e Fundamentals of Statistics and Probability Tutor-Marked Assignment

Question 1

(a) Suppose an undergraduate-admissions committee rated 400 applicants and randomly chose 12 from those in the top 15 %.

(i) Compute the probability that a person will be admitted given that he/she has the highest faculty rating among the 400 students.

(ii) Compute the probability that a person will be admitted given that he/she has the lowest faculty rating.

(b) In a production line, suppose 3 % of the products have Type A defects and 2 % of products have Type B defects. It is also known that 0.4 % of products have both types of defects. Given that a product is known to have Type A defect. Compute the probability that it has a Type B defect.

(c) Dolomite is a common rock-forming mineral and the primary component of the sedimentary rock. During mining operations, dolomite is often mixed up with shale, which is another fine-grained sedimentary rock. Miners can make use of the radioactivity features of rock to help them distinguish between shale rock zone and dolomite rock zone. Based on certain guidelines and standards, if the gamma-ray reading of a rock zone is less than 70 API units, the area is considered to be abundant in dolomite, and hence can be mined. On the other hand, if the gamma-ray reading of a rock zone exceeds 70 API units, then the area is considered to be mostly shale and therefore will not be mined. In an exploratory research study, a random set of 750 sample data is collected from a rock quarry. It is found that 480 of the samples are dolomite and 270 of the samples are shale. Of the 480 dolomite samples, 50 of them had gamma rays readings greater than 70. As for the 270 share samples, 255 of them had gamma-ray readings greater than 70. Suppose a gamma-ray reading greater than 70 is obtained at a particular depth of the rock quarry. Compute the probability that the area should be mined.

(d) The manager of a self-service carwash station found that customers take an average of 8 minutes to wash and dry their cars. Assuming that the self-service times can be modelled by an exponential distribution, compute the probability that a customer will require more than 11 minutes to complete the job.

(e) In a manufacturing process where glass products are made, bubbles do occur. When this happens, the quality of the products will be affected. Suppose based on past records, on average, 1 in every 1000 of these glass products produced has one or more bubbles.

(i) Apply a suitable exact model to compute the probability that a random sample of 5000 glass products will result in fewer than 4 products with bubbles.

(ii) Apply an approximate model to compute the probability that a random sample of 5000 glass products will result in fewer than 4 products with bubbles.

(iii) Comment on the results.

Question 2

(a) It is known that an important property of cotton fibre is its water absorbency. A random sample of 20 pieces of cotton fibre was taken and the absorbency on each piece was measured. The data is shown in Table Q2(a).

(i) Without using MS Excel or R codes, determine the following sample statistics (need to show detailed workings): Mean, Median, Mode, Standard deviation, Variance, Range, First quartile, Third quartile, Inter-quartile range, Coefficient of Variation, 90th percentile

(ii) Presents the Stem-and-Leaf plot for this data set. Comment on the distribution. If skewness statistics is to be computed, discuss its possible range of estimated values.
(b) Suppose each data in Table Q2(a) has been increased by 100, as shown in Table Q2(b). For this new data set, analyse what will happen to the following sample statistics: Mean, Mode, Range, Variance, Skewness.

Question 3

(a) It was found that the amount of fill that a machine dispenses into 200 ml bottles is uniformly distributed between 185 ml and 230 ml.

(i) Determine the average fill per bottle.

(ii) Compute the percentage of bottles with more than 210 ml.

(b) Experimental research is conducted to test the effect of an anticoagulant drug on rats. A random sample of 6 rats is used in this study. Suppose the drug manufacturer claims that 85 % of the rats will be favourably affected by the drug.

(i) Apply a suitable probability model and compute the probability that none of
the 6 experimental rats will be favourably affected by the drug.

(ii) Compute the probability that more than 2 experimental rats will be favourably affected by the drug.

(iii) Compute the probability that 2 experimental rats will not be favourably affected by the drug.

(iv) Calculate the mean number of experimental rats favourably affected by the drug.

(c) Suppose in a lottery, there are 200 prizes of $5, 20 prizes of $30, and 5 prizes of $100. Assuming that 10000 tickets are to be issued and sold, compute the fair price to pay for a ticket.

Question 4

(a) Table Q4(a) summarizes the survey data of 1929 students on students’ main reason for an application to the undergraduate’s programme that they matriculated.

(i) Given that the student is full-time, compute the probability that programming quality is the main reason for choosing.

(ii) Given that the student is part-time, compute the probability that programming cost is the main reason for choosing.

(iii) Suppose: Event A = {Student is full-time}
Event B = {Student lists programme quality as the main reason for
applying}
Show that the events A and B are dependent.

(b) Suppose at a particular traffic light junction, the traffic is green 60 % of the time, red 30 % of the time and yellow 10 % of the time. Given that a car approaches this traffic light junction once each day. Let X denotes the number of days that pass up to and including the first time the car encounters a red light. Assume that each day represents an independent trial.
(i) Apply a suitable model and compute the probability P(X = 3).

(ii) Compute the probability P(X ≤ 4).

Question 5

(a) Table Q5(a) tabulates the tensile strength of a sample of 18 bolts from the output of a thread-cutting machine.

(i) Using MS Excel, determine the following descriptive statistics: Mean, Median, Mode, Range, Standard deviation, Variance, Skewness, 85th percentile.

You are required to show a screenshot of the Excel output results.

(ii) Using R codes, determine the following descriptive statistics: Mean, Median, Standard deviation, Variance, First quartile, Third quartile, Interquartile range

You are required to show a screenshot of the R commands and output results.

(b) Suppose the number of hits on a certain website can be modelled by a Poisson distribution with a mean rate of 4 hits per minute.
(i) Compute the probability that 6 messages are received in a given minute.

(ii) Compute the probability that 8 messages are received in 1.5 minutes.

(iii) Compute the probability that fewer than 4 messages are received in a period of 30 seconds.