**INTRODUCTION TO PROBABILITY TO PROBABILITY AND STATISTICS**

**Instructions: Answer Question 1 and Any Other Two Questions.**

__QUESTION ONE (30 MARKS)__

- 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 |

- 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,
- At most 2 are red (3mks)
- At least 1 is blue (3mks)

- 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

- The mean
- Standard deviation

- Upper quartile (9mks)

- Dan computed the following statistics based on the amount x that he invested in his business, and the income y generated.
- Fit a linear regression line of y on x. (4mks)
- Estimate how much Dan would realize if he invested 5.5. (2mks)
- Given that and (5mks)

__QUESTION TWO (20 MARKS)__

- State two basic properties of a discrete probability distribution. Hence determine whether the function represent a probability distribution (4mks)
- 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)

- 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)__

- 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

- Values of A and B (5mks)
- Median (3mks)

- Mode (2mks)

- 6
^{th}Decile (3mks) - Variance (4mks)
- Given that 30% of the students failed the exam what was the cut off mark. (3mks)

__QUESTION FOUR (20 MARKS)__

- 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

- Values of x and y (5mks)
- The standard deviation (5mks)
- 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,
- Two are oranges (2mks)
- At least one is a tangerine (2mks)

- A random variable x takes on the values -3,-1, 2 and 5 with respective probabilities

. Determine

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