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UNIVERSITY EXAMINATIONS
SECOND SEMESTER 2023/2024 ACADEMIC YEAR
FIRST YEAR EXAMINATION FOR THE DEGREE OF
BACHELOR OF EDUCATION (SCIENCE/ARTS)
STAT 111: INTRODUCTION TO PROBABILITY AND STATISTICS I
STREAM: R TIME: 2 HRS
DAY: TUESDAY [ 11 .30A.M - 1.30P.M] DATE: 16 /04/
THIS QUESTION PAPER CONSISTS OF FIVE ( 5 ) PAGES
PLEASE DO NOT OPEN UNTIL THE INVIGILATOR SAYS SO.
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INSTRUCTIONS : ANSWER QUESTION ONE AND ANY OTHER TWO
QUESTION ONE [30 MARKS]
a) Explain the following terms [10 Marks] i. Convenience sampling ii. Multistage sampling iii. Sampling frame iv. Mutually exclusive events v. Random experiment b) The table below gives the number of thunderstorms reported in a particular summer month by 100 meteorological stations. i. i) Calculate the sample mean number of thunderstorms. [2 Marks] ii) Calculate the sample median number of thunderstorms. [2 Marks] iii) Comment briefly on the comparison of the mean and the median. [2 Marks] c) A random variable X has the following probability distribution X 0 1 2 3 4 5 6 7 P(x) 0 K 2k 2k 3k K^2 2k^2 7k^2 +k d) Determine; a) The value of constant k [2 Marks] b) The expectation of X [3 Marks] c) Standard deviation of X [3 Marks] e) Explain the disadvantage of using dispersion as a measure of variation [2 Marks] f) Data were collected on the time (in days) until each of 200 claims is settled by the insurer in a certain insurance portfolio. A boxplot of the data is shown below. Number of thunderstorms 0 1 2 3 4 5 Number of stations: 22 37 20 13 6 2
Vision : A University for Valued Transformation of Society Page 4 of 5 Mission: To serve students and society through research, education, scholarship, training, innovation, outreach and consultancy ii). What is the probability that at most one yellow fish gets selected? [3 Marks] iii). What is the probability that at least one yellow fish gets selected? [2 Marks] QUESTION THREE [20 MARKS] a) A survey, carried out at a major flower and gardening show, was concerned with the association between the intention to return to the show next year and the purchase of goods at this year`s show. There were 220 people interviewed and of these 101 had made a purchase; 69 of these people said they intended to return next year. Of the 119 who had not made a purchase, 68 said they intended to return next year. Suppose one of the 220 people surveyed is selected at random. Calculate the probabilities that the selected person: i) Intends to return next year, given that he/she has made a purchase [2 Marks] ii) Intends to return next year, given that he/she has not made a purchase [2 Marks] iii) Has made a purchase, given that he/she intends to return next year [ 2 Marks] b) State four disadvantages of interviewing as a method of data collection [4 Marks] c) Find the harmonic and geometric mean for the data in the frequency table below; [6 Marks] x 10 12 13 15 16 f 2 4 10 6 4 d) Given n=10, and the sum of the deviation from the mean of X and Y is 66. Find the correlation coefficient [4 Marks] QUESTION FOUR [20 MARKS] a) Explain the following terms as used in statistics [ 8 Marks] i). Regression analysis ii). Correlation analysis iii). Sampling iv). Variable b) A development engineer examined the relationship between the speed a vehicle is travelling (in miles per hour, mph), and the stopping distance (in metres, m) for a new braking system fitted to the vehicle. The following data were obtained in a series of independent tests conducted on a particular type of vehicle under identical road conditions.
Vision : A University for Valued Transformation of Society Page 5 of 5 Mission: To serve students and society through research, education, scholarship, training, innovation, outreach and consultancy c) Speed of vehicle ( x ): 10 20 30 40 50 60 70 Stopping distance (y): 5 10 23 34 40 54 75 i). Construct a scatterplot of the data, and comment on whether a linear regression is appropriate to model the relationship between the stopping distance and speed. [4 Marks] ii). Calculate the equation of the least-squares fitted regression line and interpret. [8 Marks ] QUESTION FIVE [20 MARKS] a) Find the moment coefficient of skewness and kurtosis for the data in the frequency table below; [6 Marks] X 10 12 14 18 20 F 3 5 10 5 6 b) Consider three events A , B , and C for which A and C are independent, and B and C are mutually exclusive. You are given the probabilities P ( A ) = 0.3, P ( B ) = 0.5, P ( C ) = 0. and P ( A ∩ B ) = 0.1. Find the probability that none of A , B , or C occurs. [3 Marks] c) The random variable X has probability density function Where, k is a positive constant, i) Show that k = 1.5. [2 Marks] ii) Calculate the probability P ( X > 0.25). [2 Marks] d) An analysis of the monthly wages paid to workers in two firms A and B belonging to the same industry gives the following results; Value Firm A Firm B Mean wage 52.5 47. variance 100 121 In which firm A or B is there a greater variability in individual wages? [5 Marks] e) Explain the difference between multiple and component bar charts [2 Marks]
