This week's topics covered **discrete random & continuous random variables** and **probability distributions**; including discrete, binomial, continuous and normal distributions.

### Question # 1 : Consider a population consisting of the following values, which represents the number of ice cream purchases during the academic year for each of the five housemates: 8, 14, 16, 10, 11

#### Compute the mean of this population.

> x <- c(8, 14, 16, 10, 11) > mean(x) [1] 11.8

#### Select a random sample of size 2 out of the five members. See the example used in my Power-point presentation slide # 13:

Possible Samples | x̄ |
---|---|

8, 14 | 22 / 2 = 11 |

8, 16 | 24 / 2 = 12 |

8, 10 | 18 / 2 = 9 |

8, 11 | 19 / 2 = 9.5 |

14, 16 | 30 / 2 = 15 |

14, 10 | 24 / 2 = 12 |

14, 11 | 25 / 2 = 12.5 |

16, 10 | 26 / 2 = 13 |

16, 11 | 27 / 2 = 13.5 |

10, 11 | 21 / 2 = 10.5 |

> y <- sample(x,2) > y [1] 16 8

#### Compute the mean and standard deviation of your sample.

> y <- sample(x,2) > y [1] 16 8 > mean(y) [1] 12 > sd(y) [1] 5.656854

#### Compare the mean and standard deviation of your sample to the entire population of this set (8, 14, 16, 10, 11).

x̄ = 12 s = 5.656854 μ = 11.8 σ = 3.193744

### Question # 2 : Suppose that the sample size n = 100 and the population proportion p = 0.95.

#### Does the sample proportion p have approximately a normal distribution? Explain.

- Taking 100 random samples of size 2 from the array of 5 discrete values
- Graphically displaying the results in a histogram
- Comparing this visually to the bell-curve of a normal distribution

*not*normal, due to the fact that it would be not be a continuous distribution.

#### What is the smallest value of n for which the sampling distribution of p is approximately normal?