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Math 241: Probability And Statistics Question Paper
Math 241: Probability And Statistics
Course:Bachelor Of Education Science
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
SECOND YEAR EXAMINATIONS FOR THE AWARD OF DEGREE IN BACHELOR OF EDUCATION SCIENCE
MATH 241: PROBABILITY AND STATISTICS
STREAMS: BED (SCI) Y2S2 TIME: 2 HOURS
DAY/DATE: WEDNESDAY 17/4/2013 2.30 PM – 4.30PM
INSTRUCTIONS:
Answer question ONE and any other TWO
All workings must be clearly shown
QUESTION ONE (30 MARKS – COMPULSORY)
(a) Define the following terms as used in probability and statistics;
(i) A statistic
(ii) Degrees of freedom
(iii) Expectation of a random variable. [3 Marks]
(b) (i) State the central limit theorem. [2 Marks]
(ii) Achievement test scores from all high school students in Chuka High Schools
have a mean of 60 and a variance of 64. A specific high school of n = 100 students had a mean score of 58. Calculate the probability that the sample mean is at most 58 when n = 100 and comment on it. [3 Marks]
(c) Y has the probability density function given by
?f_Y?^((y) )=[¦(2y,@0)¦ 0<y<1
elsewhere
Find the probability density function of V = -4y + 3 using the method of transformations. [4 Marks]
(d) If X is a random variable such that E (x) = 3 and E(X^2 )=13, using chebychev inequality find an upper bound for p(|x-3|=5). [5 Marks]
(e) State three important properties of a chi-square distribution. [3 Marks]
(f) The weight of a population of workers have mean 167 and standard deviation 27.
(i) If a sample of 36 workers is chosen, approximate the probability that the sample
mean of their weights lies between 163 and 170.
(ii) Repeat (i) when the sample size is 144. [10 Marks]
2. (a) State the conditions for the application of chi-square test. [5 Marks]
(b) Weights in kilograms of 10 shipments are given below:
38, 40, 45, 53, 47, 43, 55, 48, 52, 49. Use chi-square test to find out whether the variance of the distribution of weight of all shipments form which the above sample of 10 shipments was drawn is equal to 20 square kilograms.
[8 Marks]
(c) An insurance company has 25,000 automobile policy holders. If the yearly claim of a policy holder is a random variable with mean 320 and standard deviation 540, approximate the probability that the total yearly claim exceeds 8.3 million.
[7 Marks]
3. (a) To verify whether a course in accounting improved performance a similar test was
given to 12 participants both before and after the course. The original marks were recorded in alphabetical order of the participants as follows;
44, 40, 61, 52, 32, 44, 70, 41, 67, 72, 53 and 72. After the course, the marks were in the same order 53, 38, 69, 57, 46, 39, 73, 48, 73, 74, 60, and 78. Apply t – test to test whether the course was useful. [10 Marks]
(b) The screws produced by a certain machine were checked by examining number
of defectives in a sample of 12. The following table shows the distribution of 128 samples according to the number of defective items they contained.
No of defectors 0 1 2 3 4 5 6 7 Total
No. of samples 7 6 19 35 30 23 7 1 128
(i) Fit a binomial distribution and find the expected frequencies if the chance
of machine being defective is ½ [5 Marks]
(ii) Find the mean and standard deviation of the fitted distribution.
[5 Marks]
4. Certain missile components are shipped in lots of 12. Three components are selected randomly from each lot and a particular lot is accepted if none of the three components selected is defective.
(a) What is the probability that a lot will be accepted if it contains 5 defectives?
[5 Marks]
(b) What is the probability that a lot will be rejected if it contains 9 defectives?
[5 Marks]
(c) Let x be a random variable denoting the number of defectives in a sample of 3
components selected randomly from one of the above lots. If the lot contains 4 defectives, specify the probability function f (x). present the probability distribution.
(i) As a mathematical expression. [4 Marks]
(ii) In the form of a table [4 Marks]
(d) A manufacturer of watches has determined from experience that 3% of the
watches he produces are defective. If a random sample of 300 watches is examined, what is the probability that the proportion defective is between 0.02 and 0.035?. [6 Marks]
5. (a) A certain drug is claimed to be effective in curing cold. In an experiment on 500
persons with cold, half of them were given the drug and half of them were given the sugar pills. The patients reactions to the treatment are recorded in the following table.
Helped Harmed No effect Total
Drug 150 30 70 250
Sugar pills 130 40 80 250
Total 280 70 150 500
On the basis of this data can it be concluded that there is a significant difference in the effect of the drug and sugar pills? [10 Marks]
(b) A cigarette company interested in the effect of sex on the type of cigarettes
smoked and has collected the following data from a random sample of 150 persons.
Cigarette Male Female Total
A 25 30 55
B 40 15 55
C 30 10 40
Total 95 55 150
Test whether the type of cigarette smoked and sex are independent. [10 Marks]
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