#### 15. The Normal Distribution

*Question 1*

From Section 9.5 we know the height of the uniform density is 1/40. The probability to the right of a particular value *x* is

P(*X* ≥ *x*) = base × height = (190 - *x*) × (1/40).

From this we can build the following probability table:

Digits of x | ||||||||||

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

150 | 1.000 | .975 | .950 | .925 | .900 | .875 | .850 | .825 | .800 | .775 |

160 | .750 | .725 | .700 | .675 | .650 | .625 | .600 | .575 | .550 | .525 |

170 | .500 | .475 | .450 | .425 | .400 | .375 | .350 | .325 | .300 | .275 |

180 | .250 | .225 | .200 | .175 | .150 | .125 | .100 | .075 | .050 | .025 |

*Question 2*

From Section 9.5 we know the height of the triangular density is 1/20. If *x* is 170 or more then the height of the density curve at *x* is (1/20)×((190 - *x*)/20). You can check that this gives height 0 for *x*=190 and height 1/20 for *x*=170. The area to the right of *x* is then

P(*X* ≥ *x*) = (1/2) × base × height = (1/2) × (190 - *x*) × (1/20)×((190 - *x*)/20) = (1/800)×(190 - *x*)^{2}.

Similarly, for *x* ≤ 170 the area to the right is

P(*X* ≥ *x*) = 1 - (1/800)×(*x* - 150)^{2}.

From these we can build the following probability table:

Digits of x | ||||||||||

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

150 | 1.000 | .999 | .995 | .989 | .980 | .969 | .955 | .939 | .920 | .899 |

160 | .875 | .849 | .820 | .789 | .755 | .719 | .680 | .639 | .595 | .549 |

170 | .500 | .451 | .405 | .361 | .320 | .281 | .245 | .211 | .180 | .151 |

180 | .125 | .101 | .080 | .061 | .045 | .031 | .020 | .011 | .005 | .001 |

*Question 3*

Unlike the previous two questions, for Normal probabilities it is not possible to write down a formula for the area to the right of a particular *x*. Instead
we will standardize our *x* values, using μ = 167 and σ = 6.6, and then use the table of standard Normal probabilities (Table 15.1).

For example, to find P(*X* ≥ 170) we calculate *z* = (170 - 167)/6.6 = 0.45. Table 15.1 then gives P(*Z* ≥ 0.45) = .326.

Digits of x | ||||||||||

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

150 | 0.995 | 0.992 | 0.988 | 0.983 | 0.976 | 0.966 | 0.953 | 0.936 | 0.913 | 0.887 |

160 | 0.855 | 0.819 | 0.776 | 0.729 | 0.674 | 0.618 | 0.560 | 0.500 | 0.440 | 0.382 |

170 | 0.326 | 0.271 | 0.224 | 0.181 | 0.145 | 0.113 | 0.087 | 0.064 | 0.047 | 0.034 |

180 | 0.024 | 0.017 | 0.012 | 0.008 | 0.005 | 0.003 | 0.002 | 0.001 | 0.001 | 0.000 |

*Question 4*

Four boxplots are shown below. There is certainly a lot of variability in the kind of picture you get from a boxplot when the sample size is so small.