Answer all questions in Section A.
The marks for the best three answers in Section B will
be used in the assessment.
Normal and Chi-squared tables are provided at the end of the paper
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Probability and Statistics Questions and Problems, Exams of Mathematics
A series of questions and problems related to probability and statistics, including continuous and discrete random variables, hypothesis testing, poisson and normal distributions, and confidence intervals. It also includes a stem-and-leaf plot and a histogram for data analysis.
Typology: Exams
2012/2013
1 / 9
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Download Probability and Statistics Questions and Problems and more Exams Mathematics in PDF only on Docsity!
Answer all questions in Section A.
The marks for the best three answers in Section B will be used in the assessment.
Normal and Chi-squared tables are provided at the end of the paper
SECTION A
- The probability density function f(y) of a continuous random variable Y is given by
( ) (^ )^
0 otherwise. f y = K^20 y^3 −^15 y^2 if^1 ≤y≤^2
(i) Verify that K = 1/40. [2 marks] (ii) Calculate E[Y] [3 marks] (iii) Calculate the variance of Y. [3 marks]
- One group of patients with a particular disease were treated with drug A whilst another were treated with drug B. Each patient was then assessed to see if they had responded to treatment or not. The results are given in the following table.
Response to treatment Drug Yes No A 20 20 B 12 28
Test the hypothesis that there is no association between the drug given and the response to treatment. [8 marks]
- The quality control division of a food manufacturing company is concerned about the contamination of a particular product with traces of nuts. In the past it has been found that 8% of packets of the product contain traces of nuts.
A random sample of 30 packets are taken from the production line. What is the (i) expected number of packets containing nut traces? [2 marks] (ii) probability that none of the packets contain traces of nuts? [2 marks] (iii) probability that fewer than 5 packets contain traces of nuts? [4 marks]
SECTION B
- A gas company wishes to review its staffing for emergency callouts in a particular area. The company believes the number of emergency callouts per day has a Poisson distribution. They believe current staffing levels are appropriate provided the average number of emergency callouts per day is 1.
To assist its review, the company collected the following daily data over the period 01/06/
- 31/07/01 inclusive.
Number of emergency callouts
Number of days 0 6 1 20 2 18 3 10 4 7 5 0
(i) Use a Ȥ^2 test, applying Cochran’s rule, to decide whether the company needs to change its staffing levels. [7 marks] (ii) Test the hypothesis that the data are drawn from a Poisson distribution with the mean determined from the data. Apply Cochran’s rule as appropriate. [10 marks] (iii) The company considers they may need to increase the staffing levels if the probability of more than 4 callouts in a day is higher then 0.05. Do they need to increase their staffing levels? Justify your answer. [3 marks]
8(a) If X has a discrete Uniform distribution on 1, 2, …, 12, show that the mean and variance of X are 6½ and 11^11 / 12 respectively. [10 marks]
(b) A company can make up to 6 items per day. Past experience shows that the daily demand follows the probability distribution below:
Number of items sold, y 0 1 2 3 4 5 6 P(Y = y) 0.01 0.04 0.12 C 0.29 0.22 0.
Find the value of c. [2 marks]
If demand on different days is independent, what is the probability that over a two-day period, exactly 5 items will be sold? [5 marks]
The profit made by the company depends on the number of items sold as shown below:
Number of items sold, y 0 1 2 3 4 5 6 Profit, in £s -160 -80 0 80 160 240 320
Calculate the expected daily profit made by the company. [3 marks]
- The distribution of a critical dimension on auto engine crankshafts is thought to be approximately Normal with μ = 224mm and ı = 0.03mm. Crankshafts with dimensions between 223.92mm and 224.08mm are acceptable. What percentage of all crankshafts produced are acceptable? [4 marks]
A quality engineer measured a sample of 200 of these crankshafts. The table below summarises the frequency distribution of these measurements.
Dimension Frequency 223.00-223.30 2 223.30-223.50 4 223.50-223.70 9 223.70-223.80 23 223.80-223.90 32 223.90-224.00 49 224.00-224.10 34 224.10-224.20 18 224.20-224.30 12 224.30-224.50 8 224.50-224.70 6 224.70-225.00 3
Plot a histogram of these data. [8 marks]
Comment, with reasoning, on whether the histogram suggests the data are Normally distributed or not. [2 marks]
The sample mean, x, and standard deviation, s, were calculated to be 223.98 and 0. respectively. Calculate a 95% confidence interval for the mean and interpret your result. [6 marks]