# INTRODUCTION TO PROBABILITY TO PROBABILITY AND STATISTICS

INTRODUCTION TO PROBABILITY TO PROBABILITY AND STATISTICS

Instructions: Answer Question 1 and Any Other Two Questions.

QUESTION ONE (30 MARKS)

1. The table below gives a probability distribution of a discrete random variable X. given that find value of v and w (4mks)
 X 4 8 12 15 20 P(X) V 0.24 0.3 W 0.1

1. A basket contains 10 marbles 7 blue and 3 red. Three marbles are chosen at random, one at a time without replacement. Determine the probability that,
2. At most 2 are red                                                                                                                     (3mks)
3. At least 1 is blue                                                                                                                      (3mks)

1. The data below shows the number of bags of maize taken to a cooperative society deport by 100 farmers in one week
 Bags of maize 5-14 15-24 25-34 35-44 45-54 55-64 65-74 74-84 85-94 95-104 No. of farmers 18 26 30 7 1 5 4 2 5 2

Determine the following

1. The mean
2. Standard deviation
• Upper quartile                                                                                                                              (9mks)
1. Dan computed the following statistics based on the amount x that he invested in his business, and the income y generated.
2. Fit a linear regression line of y on x.                                                      (4mks)
3. Estimate how much Dan would realize if he invested 5.5.                     (2mks)
4. Given that and                                       (5mks)

QUESTION TWO (20 MARKS)

1. State two basic properties of a discrete probability distribution. Hence determine whether the function represent a probability distribution                                           (4mks)
2. The table below shows the performance of various drivers in a car rally
 Drivers name Accumulated points Distance covered Raymond 638 12560 Osteen 456 11976 Mary 729 12698 Adams 214 12213 Victor 534 11851 Ambrose 649 11620 peter 587 12321

Determine the Spearman’s correlation coefficient for this information                                                   (8mks)

1. The data in the table below shows the average attendance in percentage and the marks scored in the final examination by a random sample of 10 students
 Students A B C D E F G H J K Attendance 78 95 65 84 92 64 75 90 98 68 Mean mark 55 68 52 74 86 46 66 60 80 48

Determine the least squares regression line of the mean mark on the attendance of students (8mks)

QUESTION THREE (20 MARKS)

1. The table below shows the statistics examination marks for a class of one hundred students in their end of semester math exam.
 Marks 10-19 20-29 30-39 40-49 50-59 60-69 No. of students 4 8 A 22 48 B

Given that the mean score was 46.5 determine

1. Values of A and B                         (5mks)
2. Median                                                               (3mks)
• Mode  (2mks)
1. 6th Decile                                                                                                                                            (3mks)
2. Variance                                                                                                                  (4mks)
3. Given that 30% of the students failed the exam what was the cut off mark. (3mks)

QUESTION FOUR (20 MARKS)

1. The following are marks for 99 students in a mathematics exam.
 marks 0-10 11-20 21-30 31-40 41-50 51-60 No. of students 10 x 25 30 y 10

On later analysis, it was discovered that two class interval frequencies denoted by x and y were missing. However, the median score for this group was found to be 30 in an earlier calculation. Find

1. Values of x and y (5mks)
2. The standard deviation                              (5mks)
3. A basket contains 7 oranges and 3 tangerines. Three fruits are chosen at random, one at a time and is not replaced. Determine the probability that,
1. Two are oranges                                                                        (2mks)
2. At least one is a tangerine                                                        (2mks)
4. A random variable x takes on the values -3,-1, 2 and 5 with respective probabilities

. Determine

1. The probability distribution of x                                                                                               (4mks)
2. E(X)                                                                                                                                                      (2mks)
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