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OLLSCOIL NA hÉIREANN, GAILLIMH
NATIONAL UNIVERSITY OF IRELAND, GALWAY
SPRING EXAMINATIONS 2009
MODULE CODE: MA 419/BC
MODULE INSTANCES:
1AM1 - Higher Diploma in Applied Science (Microbiology) 3EV2 - Bachelor of Science (Environmental Science) (Hons.) 1AS1 - H.Dip. Applied Science (Analytical Biochemistry/Chemistry) 1CB1 - Master of Science (Analytical Biochemistry/Chemistry)
MODULE: STATISTICS
External Examiners: Prof. E. M. Scott Internal Examiner: Dr. J.N. Sheahan
INSTRUCTIONS: Answer the ten questions in PART A (30 marks) and two of
the four questions B1, B2. B3 and B4 in PART B ( 35 marks each).
Tables of the standard normal and t distributions are attached.
DURATION : Three hours
The questions commence on the next page.
PART A
[Multiple choice. 30 marks] In each of questions A1. through A10. below, write down one choice of answer. For example, if in A1. below you think (A) is the answer, you would write in your answer book A1. (A).
A1. A lecturer is performing a statistical test concerning the effectiveness of a new teaching technique. If he committed a Type II error by erroneously concluding that the technique is not effective, what were the alternatives he was testing? (A) H 0 : the technique is not effective, H 1 : the technique is effective, (B) H 0 : the technique is effective, H 1 : the technique is not effective.
A2. Suppose you have one line of 6 numbers for the next draw of the Lotto 456 , in which six numbers
are drawn at random from the 45 numbers {1, 2, ..., 45}, and then a seventh number (called the bonus) is drawn. What is the probability that you will get ‘five and the bonus’, that is, five of the first six numbers drawn, and the bonus number?
(A)
(B)
(C)
(D)^6
6 45
(^5 )
A3. Among the 100 students in final year of a Science degree, there were 60 males and 40 females. Suppose that 30 of the males and 20 of the women take Chemistry as a subject. What is the probability that a randomly sampled student is a male and studies Chemistry? (A) 0.2 (B) 0.3 (C) 0.4 (D) 0.5 (E) 0.6 (F) 0.
A4. From a collection of 2 types of bacteria and 4 types of viruses, a microbiologist can choose 3
microorganisms in 6 3 20 ways. If she chooses the three microorganisms at random, what is the probability that the selection will have at least one of each of the two types of microorganisms? (A) 0.8 (B) 0.7 (C) 0.6 (D) 0.5 (E) 0.4 (F) 0.
A5. Student A will take a random sample of size 900 from an infinite population. Student B will take a random sample of size n from the same population. How large should n be if it is desired that the standard deviation of the mean X (^) A of A ’s sample be three times smaller than the standard deviation of the mean X (^) B of B ’s sample?
(A) 16 (B) 30 (C) 50 (D) 100 (E) 150 (F) 400.
A6. Suppose that a population of marks of students has a normal distribution with mean 60 and standard deviation 10. Let a P mark of a random student falls between 50 and 70 and let b P the mean mark for a random sample of 100 students will be between 58 and 62. Then
(A) a 0. 9544 and b 0. 9544 (B) a 0. 6826 and b 0. 9544
(C) a 0. 9544 and b 0. 6826 (D) a 0. 6826 and b 0. 6826
(E) a 0. 0013 and b 0. 0228 (F) a 0. 0228 and b 0. 0013.
PART B
Answer two of B1, B2, B3 and B
B1.
(a) What are the median and the standard deviation of the set of numbers {3, 4, 5}? [ 1 2 3 marks] (b) Write down the set of three numbers which have a mean equal to 4, and have a (unique) mode equal to 3. [ 3 marks] (c) The letters of the word STATISTICS are arranged in all possible ways. If an arrangement is chosen at random, calculate the probability that the three vowels will be next to each other. [ 6 marks] (d) A and B are events in a sample space satisfying P A 0. 5, P B 0. 2 and P A B 0. 15. [ Note: Recall that P A B means the probability that A and B occur simultaneously.] Are A and B independent? Justify your answer, as usual. [ 3 marks] (e) In a certain company, 60% of replacement parts are made by the company itself, while the remaining 40% come from external sources. Suppose that 5% of the parts made by the company are defective while 10% of the parts made by external sources are defective. A part is selected at random and found to be defective. Use Bayes’ formula to find the probability that this part is company made. [ 8 marks] (f) In a certain classroom there are 10 people, of whom 6 are males. (i) If three people are selected at random from the room, find the probability that exactly two of them are males. (ii) What is the answer in (i) above if sampling is with replacement? [4 4 8 marks] (g) Assume that the binomial distribution with p P a randomly selected child is male 0. 5, is an appropriate model for the distribution of X the number of male children in a randomly selected family that has n 5 children. Calculate the probability that a family with 5 children will contain exactly three male children. [ 4 marks]
B2.
Assume that the time X that a random NUI, Galway student spends on a computer per week has a normal distribution with mean 7 hours and standard deviation 2 hours. (i) If a student is selected at random, find the probability that the length of time that he/she spends on a computer in a week is between 7 and 10 hours. [ 9 marks] (ii) If 15.87% of students spend less than a hours on computers in the week, find a. [ 9 marks] (iii) Find the probability that the mean time, X , spent on computers in a random week by n 64 randomly selected students exceeds 7.5 hours. [ 9 marks] (iv) Carefully state the Central Limit Theorem and then answer the following: In calculating the probability in (iii) above, was it necessary to know that the underlying random variable X is normally distributed? Briefly justify your answer. [ 6 2 8 marks]
B3.
(a) Give an example of the role of probability in statistical inference. Ensure that you are very precise in your definitions of the population(s) and the possible parameter(s) about which you want to make an inference. Describe clearly the sample(s) taken from the population(s), give your probability calculation, and finally state clearly your inference with justification. [7 marks]
(b) Assume that the population of weights of people (in kgs.) has an approximately normal distribution with unknown mean and standard deviation 10. 0. This question concerns the z -test for testing the null hypothesis H 0 : 60. 0 against the alternative hypothesis H 1 : 60. 0 using 0. 05. Suppose that a random sample of n 25 people was chosen, and their weights analyzed by a statistical software package. Part of the output is shown in the following computer printout. Test of H 0 : 60. 0 versus H 1 : 60. 0.
n Mean x z -value 25 58. 0 1. 0
Helpful note: The test rejects H 0 if z (^) OBS z (where z (^) OBS x^ ^60 / n
or equivalently if
x 60 z n. That is, the critical region of the test is x : x 60 z n.
(i) The printout give the z -value (i.e. z (^) OBS to be 1. 0. Show that this value is indeed correct by calculating the z -value yourself.. [2 marks] (ii) Based on the printout, should H 0 be rejected? Give briefly a reason for your answer, based on either x , or the z -value (i.e. z (^) OBS ) [2 marks] (iii) Use the printout to calculate the p-value of the test, and then state – with justification – whether or not H 0 should be rejected [8 marks] (iv) Calculate the power of the above z -test when 54. 71. [8 marks] (v) Now ignore the computer output above. Find the appropriate critical region so that the test would have a significance level of 0. 0013 based on a random sample of n 100. [8 marks]
Standard Normal Distribution
The table entries represent the proportion p of area under the N(0,1) curve above the indicated values of z. (Example 0.3050 of the area is above z = 0.51.
z
Student's t Distribution For various degrees of freedom ( df) , the tabled entries represent the critical values of t above which a specified portion p of the t -distribution falls. (Example: For df = 5, a t of 2.0150 is surpassed by 0.05 or 5% of the total distribution). t
- z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.085 0. Second decimal place of z
- 0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4661 0.
- 0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4266 0.
- 0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3878 0.
- 0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3501 0.
- 0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3138 0.
- 0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2793 0.
- 0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2467 0.
- 0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2162 0.
- 0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1881 0.
- 0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1623 0. - 1 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1390 0.
- 1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1180 0.
- 1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.0994 0.
- 1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0830 0.
- 1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0688 0.
- 1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0565 0.
- 1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0460 0.
- 1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0371 0.
- 1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0297 0.
- 1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0236 0. - 2 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0185 0.
- 2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0144 0.
- 2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0112 0.
- 2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0085 0.
- 2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0065 0.
- 2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.
- 2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0036 0.
- 2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.
- 2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.
- 2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.
- df 0.1 0.05 0.025 0.01 0. p (One-tailed probabilities) - 1 3.0777 6.3138 12.7062 31.8205 63. - 2 1.8856 2.9200 4.3027 6.9646 9. - 3 1.6377 2.3534 3.1824 4.5407 5. - 4 1.5332 2.1318 2.7764 3.7470 4. - 5 1.4759 2.0150 2.5706 3.3649 4. - 6 1.4398 1.9432 2.4469 3.1427 3. - 7 1.4149 1.8946 2.3646 2.9980 3. - 8 1.3968 1.8595 2.3060 2.8965 3. - 9 1.3830 1.8331 2.2622 2.8214 3.
- 10 1.3722 1.8125 2.2281 2.7638 3.
- 11 1.3634 1.7959 2.2010 2.7181 3.
- 12 1.3562 1.7823 2.1788 2.6810 3.
- 13 1.3502 1.7709 2.1604 2.6503 3.
- 14 1.3450 1.7613 2.1448 2.6245 2.
- 15 1.3406 1.7531 2.1315 2.6025 2.
- 16 1.3368 1.7459 2.1199 2.5835 2.
- 17 1.3334 1.7396 2.1098 2.5669 2.
- 18 1.3304 1.7341 2.1009 2.5524 2.
- 19 1.3277 1.7291 2.0930 2.5395 2.
- 20 1.3253 1.7247 2.0860 2.5280 2.
- 21 1.3232 1.7207 2.0796 2.5176 2.
- 22 1.3212 1.7171 2.0739 2.5083 2.
- 23 1.3195 1.7139 2.0687 2.4999 2.
- 24 1.3178 1.7109 2.0639 2.4922 2.
- 25 1.3163 1.7081 2.0595 2.4851 2.
- 26 1.3150 1.7056 2.0555 2.4786 2.
- 27 1.3137 1.7033 2.0518 2.4727 2.
- 28 1.3125 1.7011 2.0484 2.4671 2.
- 29 1.3114 1.6991 2.0452 2.4620 2.
- 30 1.3104 1.6973 2.0423 2.4573 2.
- 40 1.3031 1.6839 2.0211 2.4233 2.
- 60 1.2958 1.6706 2.0003 2.3901 2.
- 120 1.2886 1.6577 1.9799 2.3578 2. - ∞ 1.2817 1.6452 1.9604 2.3271 2.
